#6685307

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#37785	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1901 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)(7,10)(8,9)(11,12)(13,14,19,15)(16,24,26,28)(17,21,18,23)(20,25,22,27);(1,2)(3,4)(6,9)(7,11)(13,15,19,14)(16,22,26,20)(17,21,18,23)(24,27,28,25);(2,4)(13,16,17,25,19,26,18,27)(14,20,23,28,15,22,21,24);(6,9)(10,12)(13,17,19,18)(14,21,15,23)(16,25,26,27)(20,24,22,28);(2,4)(5,7,8,11)(6,10,9,12)(16,26)(20,22)(24,28)(25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592503);