#6994337

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2226 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)(15,24)(16,21)(17,19)(18,23)(20,26)(22,32)(25,37)(27,36)(28,34)(29,35)(30,39)(31,38)(33,42)(40,44)(41,43);(5,6)(7,11)(8,10)(9,12)(14,19)(16,28)(17,32)(20,33)(22,23)(24,36)(25,40)(27,42)(29,41)(30,31)(34,39)(43,44);(2,4)(5,7)(6,9)(8,12)(10,11)(13,15,18,26)(14,20,23,24)(16,29,31,25)(17,27,32,42)(19,33,22,36)(21,37,38,35)(28,41,30,40)(34,44,39,43);(7,12)(9,11)(13,16,18,31)(14,21,23,38)(15,25,26,29)(17,28,32,30)(19,34,22,39)(20,35,24,37)(27,40,42,41)(33,43,36,44);(6,10)(9,11)(15,26)(20,24)(25,29)(27,42)(33,36)(35,37)(40,41)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6901533);