#6766588

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1592 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)(17,18,22,27)(19,23,30,39)(20,25,31,41)(21,33,35,26)(24,34,40,32)(28,37,42,48)(29,44,46,38)(36,45,47,43);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,19,22,30)(18,23,27,39)(20,28,31,42)(21,29,35,46)(24,36,40,47)(25,37,41,48)(26,38,33,44)(32,43,34,45);(10,13)(14,16)(17,20)(18,24)(19,28)(21,34)(22,31)(23,36)(25,33)(26,41)(27,40)(29,45)(30,42)(32,35)(37,44)(38,48)(39,47)(43,46);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,28)(21,32)(22,31)(23,37)(24,33)(26,40)(27,41)(29,43)(30,42)(34,35)(36,44)(38,47)(39,48)(45,46);(3,8)(5,7)(10,13)(14,16)(20,31)(24,40)(25,41)(28,42)(32,34)(36,47)(37,48)(43,45)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,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, 6673784);