#6570658

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#30270	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2113 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)(5,6,11,10)(7,8,9,12)(13,14,19,23)(15,24,28,41)(16,21,32,39)(17,22,18,33)(20,26,35,27)(25,40,29,44)(30,37,31,38)(34,43,36,42);(5,7,11,9)(6,12,10,8)(13,15,17,26,19,28,18,27)(14,20,33,41,23,35,22,24)(16,29,31,42,32,25,30,43)(21,36,37,40,39,34,38,44);(1,2,3,4)(5,8,11,12)(6,7,10,9)(13,16,19,32)(14,21,23,39)(15,25,28,29)(17,31,18,30)(20,34,35,36)(22,38,33,37)(24,40,41,44)(26,43,27,42);(5,9,11,7)(6,12,10,8)(13,17,19,18)(14,22,23,33)(15,26,28,27)(16,30,32,31)(20,24,35,41)(21,37,39,38)(25,42,29,43)(34,44,36,40);(5,10,11,6)(7,8,9,12)(13,17,19,18)(14,22,23,33)(15,26,28,27)(16,31,32,30)(20,24,35,41)(21,38,39,37)(25,43,29,42)(34,40,36,44)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=gi,d^2=h,e^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^c=di,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6477854);