#6756768

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 ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1455 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)(3,6)(4,8)(5,7)(17,18,22,26)(19,23,28,37)(20,25,32,42)(21,24,34,27)(29,36,31,40)(30,38,43,35)(33,39,46,41)(44,48,45,47);(1,3,6,2)(4,8,5,7)(9,10)(11,15)(12,14)(13,16)(17,19,22,28)(18,23,26,37)(20,31,32,29)(21,30,34,43)(24,38,27,35)(25,40,42,36)(33,45,46,44)(39,47,41,48);(9,11,12,16)(10,13,14,15)(18,24)(19,28)(21,34)(23,35)(25,39)(26,27)(29,31)(33,46)(36,47)(37,38)(40,48)(41,42);(1,4,6,5)(2,7,3,8)(17,20)(18,25)(19,29)(21,33)(22,32)(23,36)(24,39)(26,42)(27,41)(28,31)(30,44)(34,46)(35,47)(37,40)(38,48)(43,45);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(18,26)(19,28)(20,32)(24,27)(30,43)(33,46)(36,40)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=hi,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6663964);