#6630118

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2102 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)(5,6,8,10)(7,11,12,9)(13,14,17,19)(15,20,24,32)(16,26,27,36)(18,30,29,22)(21,31,33,28)(23,37,38,44)(25,41,40,35)(34,42,43,39);(5,7,8,12)(6,9,10,11)(14,19)(20,32)(21,33)(22,30)(26,36)(34,43)(35,41)(37,44);(1,2,3,4)(13,15)(14,20)(16,23)(17,24)(18,25)(19,32)(21,34)(22,35)(26,37)(27,38)(28,39)(29,40)(30,41)(31,42)(33,43)(36,44);(13,16,17,27)(14,21,19,33)(15,23,24,38)(18,31,29,28)(20,34,32,43)(22,36,30,26)(25,42,40,39)(35,44,41,37);(15,24)(20,32)(23,38)(25,40)(34,43)(35,41)(37,44)(39,42)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,e^c=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, 6537314);