#7557278

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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1757 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,d^2=g,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,e^b=ei,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, 7464474);