#6994426

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	#42908	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2168 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=i,e^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=ehi,f^b=fi,d^c=dhi,e^c=ei,f^d=fi

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, 6901622);