#7062938

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
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1882 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)(14,20)(17,18)(21,35)(22,28)(23,33)(24,37)(26,32)(30,34)(36,44)(38,41)(39,40)(42,43);(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,24,27,21)(16,28,31,39)(17,20,18,33)(22,30,40,34)(25,41,29,44)(26,35,32,37)(36,42,38,43);(2,4)(5,7,8,12)(6,9,10,11)(13,15,19,27)(14,21,23,24)(16,29,31,25)(17,32,18,26)(20,35,33,37)(22,38,40,36)(28,41,39,44)(30,42,34,43);(7,12)(9,11)(13,16,19,31)(14,22,23,40)(15,25,27,29)(17,30,18,34)(20,28,33,39)(21,36,24,38)(26,42,32,43)(35,44,37,41);(6,10)(9,11)(13,17,19,18)(14,20,23,33)(15,26,27,32)(16,30,31,34)(21,37,24,35)(22,28,40,39)(25,42,29,43)(36,41,38,44)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,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=dh,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, 6970134);