#6676453

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#30509	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1951 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6583649);