#7613825

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1901 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)(5,6)(7,11)(8,9)(10,12)(13,14,18,22)(15,19,24,30)(16,20,28,34)(17,21,27,33)(23,29,37,41)(25,31,36,40)(26,32,38,42)(35,39,43,44);(2,3)(6,9)(10,11)(16,26)(17,27)(20,32)(21,33)(23,35)(25,36)(28,38)(29,39)(31,40)(34,42)(37,43)(41,44);(1,2,4,3)(13,15,18,24)(14,19,22,30)(16,23,28,37)(17,25,27,36)(20,29,34,41)(21,31,33,40)(26,35,38,43)(32,39,42,44);(1,3,4,2)(13,16,27,38,18,28,17,26)(14,20,33,42,22,34,21,32)(15,23,36,43,24,37,25,35)(19,29,40,44,30,41,31,39);(5,7,8,12)(6,10,9,11)(15,24)(16,28)(19,30)(20,34)(25,36)(26,38)(31,40)(32,42)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521021);