#6463774

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#533 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1960 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gi,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=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, 6370970);