#6684835

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#24425	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6,8,9)(7,11,12,10)(13,14,17,20)(15,22,24,27)(16,19,25,21)(18,26,28,23);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,21)(17,20)(19,25)(22,23)(24,28)(26,27);(2,4)(6,9)(10,11)(13,15,16,23,17,24,25,26)(14,18,21,22,20,28,19,27);(5,7,8,12)(6,10,9,11)(13,14)(15,18)(16,21)(17,20)(19,25)(22,23)(24,28)(26,27);(2,4)(13,16,17,25)(14,19,20,21)(15,23,24,26)(18,27,28,22)

Polycyclic presentation:

2^9|a^2=f,c^2=fgi,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592031);