#7062244

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45092	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1870 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6969440);