#6598251

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1848 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:

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

Polycyclic presentation:

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

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, 6505447);