#6585500

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#36028	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1871 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492696);