#53355

order	256
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

615 conjugacy classes of subgroups; only the 395 normal subgroups are shown.

normal subgroups only

This group has 615 conjugacy classes of subgroups — too many to draw. Only its 395 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(3,9)(7,12)(10,15)(14,16)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(33,34)(35,41)(36,44)(37,40)(38,43)(39,45)(42,46)(47,48);(2,6)(4,11)(7,12)(10,15)(21,29)(24,28)(26,32)(30,31)(33,34,37,40)(35,41,43,38)(36,44,45,39)(42,46,48,47);(1,2,5,6)(3,7,9,12)(4,8,11,13)(10,14,15,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(33,34)(35,38)(36,39)(37,40)(41,43)(42,46)(44,45)(47,48);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,47)(45,48);(1,4)(2,8)(3,10)(5,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,36,37,45)(34,39,40,44)(35,42,43,48)(38,46,41,47)

Polycyclic presentation:

2^8|a^2=g,b^2=f,c^2=fgh,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#53355⋊C₃
maximal subgroup of	#97315, #179425, #179426, …
maximal normal subgroup of	#97315, #179425, #179426, …
maximal quotient of	#606838, #607220, #607230, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 46343);