#7062890

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

1839 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)(15,25)(17,18)(21,36)(22,29)(23,24)(26,35)(27,30)(31,34)(37,41)(38,44)(39,40);(5,6)(7,11)(8,10)(9,12)(13,14,19,24)(15,26,25,21)(16,29,32,39)(17,20,18,23)(22,31,40,34)(27,41,30,44)(28,35,33,36)(37,42,38,43);(2,4)(5,7,8,12)(6,9,10,11)(13,15,19,25)(14,21,24,26)(16,30,32,27)(17,33,18,28)(20,35,23,36)(22,38,40,37)(29,41,39,44)(31,42,34,43);(7,12)(9,11)(13,16,19,32)(14,22,24,40)(15,27,25,30)(17,31,18,34)(20,29,23,39)(21,37,26,38)(28,42,33,43)(35,44,36,41);(6,10)(9,11)(13,17,19,18)(14,23,24,20)(15,28,25,33)(16,31,32,34)(21,35,26,36)(22,39,40,29)(27,42,30,43)(37,44,38,41)

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=eh,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, 6970086);