#6996733

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#45093	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2338 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6903929);