#7062954

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×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1843 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)(14,20)(15,24)(17,18)(21,36)(22,28)(23,33)(25,35)(26,29)(30,34)(37,41)(38,44)(39,40);(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,25,24,21)(16,28,31,39)(17,20,18,33)(22,30,40,34)(26,41,29,44)(27,35,32,36)(37,42,38,43);(2,4)(5,7,8,12)(6,9,10,11)(13,15,19,24)(14,21,23,25)(16,29,31,26)(17,32,18,27)(20,35,33,36)(22,38,40,37)(28,41,39,44)(30,42,34,43);(7,12)(9,11)(13,16,19,31)(14,22,23,40)(15,26,24,29)(17,30,18,34)(20,28,33,39)(21,37,25,38)(27,42,32,43)(35,44,36,41);(6,10)(9,11)(13,17,19,18)(14,20,23,33)(15,27,24,32)(16,30,31,34)(21,36,25,35)(22,28,40,39)(26,42,29,43)(37,41,38,44)

Polycyclic presentation:

2^9|a^2=gi,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, 6970150);