#58975

order	256
nilpotency class	2
isoclinism	isoclinic class of #53390
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:

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

Polycyclic presentation:

2^8|a^2=h,b^2=f,c^2=fgh,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#58975⋊C₃
maximal subgroup of	C₂×#58975, #8105238, #8105311, …
maximal normal subgroup of	C₂×#58975, #8105238, #8105311, …
maximal quotient of	#586102, #588316, #590518, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 51963);