#53383

order	256
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	Aut(G) order 98304
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.

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

Polycyclic presentation:

2^8|a^2=fg,b^2=f,c^2=fgh,d^2=f,e^2=fg,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	#53383⋊C₃
maximal subgroup of	C₂×#53383, #8105918, #8105925, …
maximal normal subgroup of	C₂×#53383, #8105918, #8105925, …
maximal quotient of	#606836, #607185, #607186, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 46371);