#6584600

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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 ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1926 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)(8,12)(9,10)(13,14,19,24)(15,20,16,21)(17,22,18,23)(25,27,26,28);(6,10)(11,12)(13,15,19,16)(14,20,24,21)(17,26,18,25)(22,28,23,27);(1,2,3,4)(5,7,9,8)(6,11,10,12)(15,25)(16,26)(17,18)(20,27)(21,28)(22,23);(5,8,9,7)(6,12,10,11)(13,16,19,15)(14,21,24,20)(17,25,18,26)(22,27,23,28);(13,17)(14,22)(18,19)(23,24)(25,26)(27,28)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,e^c=ei,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, 6491796);