#6500437

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500434
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2732 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:

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

Polycyclic presentation:

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

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, 6407633);