#7565226

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
prime radical	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1815 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)(7,11)(8,12)(9,10)(13,14)(15,25)(16,30)(17,20)(18,24)(19,23)(21,29)(22,34)(26,35)(27,39)(28,38)(31,36)(32,37)(33,40)(41,43)(42,44);(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)(5,7,9,8)(6,11,10,12)(13,16,19,31,17,33,18,34)(14,22,24,40,20,36,23,30)(15,27,26,42,28,32,29,43)(21,37,38,44,35,39,25,41);(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|c^2=fg,d^2=fg,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, 7472422);