#6746113

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1710 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=fi,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, 6653309);