#6630226

order	512
nilpotency class	3
isoclinism	isoclinic class of #6628851
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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2152 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537422);