#6630150

order	512
nilpotency class	3
isoclinism	isoclinic class of #6629011
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2130 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)(5,6,8,10)(7,11,12,9)(13,14,17,19)(15,23,25,36)(16,21,27,32)(18,30,29,22)(20,33,31,26)(24,37,38,44)(28,41,40,35)(34,43,42,39);(5,7,8,12)(6,9,10,11)(14,19)(20,31)(21,32)(22,30)(23,36)(34,42)(35,41)(37,44);(13,15,18,26,17,25,29,33)(14,20,22,23,19,31,30,36)(16,24,28,39,27,38,40,43)(21,34,35,37,32,42,41,44);(2,4)(13,16)(14,21)(15,24)(17,27)(18,28)(19,32)(20,34)(22,35)(23,37)(25,38)(26,39)(29,40)(30,41)(31,42)(33,43)(36,44);(16,27)(21,32)(24,38)(28,40)(34,42)(35,41)(37,44)(39,43)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537346);