#6858817

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1479 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)(17,18,21,26)(19,28,30,23)(20,32,33,25)(22,24,35,27)(29,37,42,41)(31,38,44,36)(34,40,45,39)(43,46,48,47);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,21,30)(18,23,26,28)(20,29,33,42)(22,31,35,44)(24,36,27,38)(25,37,32,41)(34,43,45,48)(39,46,40,47);(10,14)(15,16)(18,24)(22,35)(23,36)(25,39)(26,27)(28,38)(31,44)(32,40)(34,45)(37,46)(41,47)(43,48);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,37)(24,39)(26,32)(27,40)(28,41)(30,42)(31,43)(35,45)(36,46)(38,47)(44,48);(3,8)(5,7)(9,12,13,11)(10,15,14,16)(20,33)(25,32)(29,42)(34,45)(37,41)(39,40)(43,48)(46,47)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6766013);