#6867393

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#31046	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1498 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,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, 6774589);