#7564610

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1920 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7471806);