#6606629

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1830 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)(13,14,19,21,18,15,17,22)(16,20,24,28,25,23,26,27);(5,6,8,9)(7,10,11,12)(13,15,18,14)(16,23,25,20)(17,22,19,21)(24,28,26,27);(1,2,3,4)(6,9)(7,11)(13,16,18,25)(14,20,15,23)(17,26,19,24)(21,28,22,27);(5,7)(6,10)(8,11)(9,12)(13,17,18,19)(14,21,15,22)(16,24,25,26)(20,27,23,28);(6,9)(7,11)(16,25)(20,23)(24,26)(27,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513825);