#6464016

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1972 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di

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, 6371212);