#7527520

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	#45236	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1996 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=f,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,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, 7434716);