#6683243

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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	#24397	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2030 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)(7,11)(8,10)(9,12)(13,14,18,22)(15,21,17,20)(16,25,27,24)(19,28,26,23);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,23)(17,18)(19,24)(21,22)(25,26)(27,28);(2,4)(13,16,26,20,18,27,19,21)(14,17,28,24,22,15,23,25);(6,10)(9,11)(13,17,18,15)(14,21,22,20)(16,23,27,28)(19,25,26,24);(2,4)(7,12)(9,11)(15,17)(16,27)(20,21)(24,25)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,d^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,d^c=di,e^c=ei,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, 6590439);