#6745797

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2301 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6652993);