#7002899

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#42796	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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)(3,7,8,5)(9,10)(11,13)(12,14)(15,16)(17,18,23,37,21,24,22,27)(19,28,36,25,29,43,31,39)(20,32,38,42,33,26,35,47)(30,45,48,41,34,44,46,40);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,14)(13,16)(18,24)(19,29)(20,33)(27,37)(31,36)(35,38)(40,41)(44,45);(9,11)(10,13)(12,16)(14,15)(17,19,21,29)(18,25,24,39)(20,34,33,30)(22,36,23,31)(26,41,32,40)(27,43,37,28)(35,46,38,48)(42,44,47,45);(3,8)(5,7)(11,16)(13,15)(17,20)(18,26)(19,30)(21,33)(22,35)(23,38)(24,32)(25,40)(27,42)(28,44)(29,34)(31,46)(36,48)(37,47)(39,41)(43,45);(10,14)(13,15)(18,24)(20,33)(25,39)(27,37)(28,43)(30,34)(35,38)(46,48)

Polycyclic presentation:

2^9|a^2=fg,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,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, 6910095);