#7143826

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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
prime radical	#45332	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2201 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)(7,11)(10,12)(13,14,19,17)(15,23,27,26)(16,21,30,33)(18,34,24,22)(20,35,32,28)(25,39,29,42)(31,44,40,38)(36,43,37,41);(6,9)(7,11)(13,15,24,35,19,27,18,28)(14,20,34,26,17,32,22,23)(16,29,40,41,30,25,31,43)(21,37,44,39,33,36,38,42);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,25)(17,33)(18,31)(19,30)(20,36)(22,38)(23,39)(24,40)(26,42)(27,29)(28,41)(32,37)(34,44)(35,43);(6,9)(10,12)(13,17,19,14)(15,26,27,23)(16,21,30,33)(18,22,24,34)(20,28,32,35)(25,39,29,42)(31,44,40,38)(36,43,37,41);(5,7,8,11)(6,10,9,12)(15,27)(16,30)(20,32)(21,33)(23,26)(28,35)(31,40)(38,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,d^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=dfi,e^b=eh,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, 7051022);