#6671304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1811 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)(15,17)(16,22)(18,20)(19,25)(21,23)(24,26)(27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,24)(18,21)(19,25)(20,23)(22,26)(27,28);(2,4)(6,9)(10,11)(13,16,18,26)(14,19,20,28)(15,22,21,24)(17,27,23,25);(5,7,8,12)(6,10,9,11)(13,14)(15,23)(16,19)(17,21)(18,20)(22,25)(24,27)(26,28);(2,4)(14,20)(15,21)(19,28)(22,24)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^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, 6578500);