#6464019

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437758
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1900 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(2,3)(13,14,19,33,18,23,17,22)(15,24,32,20,26,39,27,36)(16,28,34,38,30,21,31,43)(25,41,44,37,29,40,42,35);(1,2,4,3)(5,6)(7,11)(8,9)(10,12)(13,15,18,26)(14,20,23,36)(16,29,30,25)(17,32,19,27)(21,37,28,35)(22,39,33,24)(31,42,34,44)(38,40,43,41);(1,3,4,2)(6,9)(10,11)(13,16,18,30)(14,21,23,28)(15,25,26,29)(17,31,19,34)(20,35,36,37)(22,38,33,43)(24,40,39,41)(27,42,32,44);(5,7,8,12)(6,10,9,11)(15,26)(20,36)(24,39)(25,29)(27,32)(35,37)(40,41)(42,44);(16,30)(21,28)(25,29)(31,34)(35,37)(38,43)(40,41)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6371215);