#7592948

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 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)(5,6,8,10)(7,9,11,12)(13,14,19,24)(15,25,28,22)(16,21,29,36)(17,20,32,35)(18,33,26,23)(27,39,41,37)(30,34,42,44)(31,43,40,38);(2,3)(5,7)(6,9)(8,11)(10,12)(13,15,26,32,19,28,18,17)(14,20,33,22,24,35,23,25)(16,27,40,42,29,41,31,30)(21,34,43,37,36,44,38,39);(1,2,4,3)(13,16)(14,21)(15,27)(17,30)(18,31)(19,29)(20,34)(22,37)(23,38)(24,36)(25,39)(26,40)(28,41)(32,42)(33,43)(35,44);(1,3,4,2)(7,11)(9,12)(15,28)(16,29)(17,32)(20,35)(21,36)(22,25)(31,40)(38,43);(13,17,18,28,19,32,26,15)(14,22,23,20,24,25,33,35)(16,30,31,41,29,42,40,27)(21,37,38,34,36,39,43,44)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=g,d^2=g,e^2=f,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, 7500144);