#6606896

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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	#37683	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2303 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)(7,10)(8,11)(9,12)(13,14,19,23)(15,24,27,34)(16,21,29,37)(17,22,18,32)(20,31,35,26)(25,38,40,43)(28,36,30,42)(33,41,44,39);(5,7)(6,9)(8,12)(10,11)(13,15,18,26,19,27,17,31)(14,20,22,24,23,35,32,34)(16,25,30,39,29,40,28,41)(21,33,36,38,37,44,42,43);(1,2,3,4)(7,12)(9,10)(13,16,19,29)(14,21,23,37)(15,25,27,40)(17,28,18,30)(20,33,35,44)(22,36,32,42)(24,38,34,43)(26,39,31,41);(5,7)(6,10)(8,12)(9,11)(13,17,19,18)(14,22,23,32)(15,26,27,31)(16,28,29,30)(20,34,35,24)(21,36,37,42)(25,39,40,41)(33,43,44,38);(7,12)(9,10)(15,27)(16,29)(20,35)(21,37)(24,34)(26,31)(28,30)(36,42)

Polycyclic presentation:

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

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, 6514092);