#6464022

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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

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,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, 6371218);