#7143788

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2257 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)(7,11)(10,12)(13,14,18,22)(15,23,26,34)(16,20,29,37)(17,31,32,21)(19,25,35,27)(24,39,28,43)(30,44,40,38)(33,42,36,41);(6,9)(7,11)(13,15,17,27,18,26,32,25)(14,19,21,23,22,35,31,34)(16,28,30,42,29,24,40,41)(20,36,38,43,37,33,44,39);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,29)(14,20,22,37)(15,24,26,28)(17,30,32,40)(19,33,35,36)(21,38,31,44)(23,39,34,43)(25,42,27,41);(6,9)(10,12)(15,25)(16,29)(17,32)(19,34)(20,37)(21,31)(23,35)(24,41)(26,27)(28,42)(33,39)(36,43);(5,7,8,11)(6,10,9,12)(15,26)(16,29)(19,35)(20,37)(23,34)(25,27)(30,40)(38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,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, 7050984);