#7611816

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	#45265	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1999 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)(7,9)(8,10)(11,12)(13,14,19,23)(15,24,27,35)(16,21,30,40)(17,22,18,33)(20,31,36,26)(25,41,29,44)(28,37,32,39)(34,42,38,43);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15,17,31,19,27,18,26)(14,20,33,35,23,36,22,24)(16,25,32,42,30,29,28,43)(21,34,37,44,40,38,39,41);(6,10)(7,11)(13,16)(14,21)(15,25)(17,32)(18,28)(19,30)(20,34)(22,39)(23,40)(24,41)(26,43)(27,29)(31,42)(33,37)(35,44)(36,38);(1,2,3,4)(6,10)(9,12)(13,17,19,18)(14,22,23,33)(15,26,27,31)(16,28,30,32)(20,35,36,24)(21,37,40,39)(25,42,29,43)(34,41,38,44);(13,15,17,31,19,27,18,26)(14,20,33,35,23,36,22,24)(16,29,32,43,30,25,28,42)(21,38,37,41,40,34,39,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^2=h,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^c=ei,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, 7519012);