#7143765

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	#45248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2203 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)(15,23,28,26)(16,21,30,29)(18,33,24,22)(20,34,32,27)(25,37,40,39)(31,43,38,36)(35,44,42,41);(6,9)(7,11)(13,15,24,34,19,28,18,27)(14,20,33,26,17,32,22,23)(16,25,38,44,30,40,31,41)(21,35,43,39,29,42,36,37);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,30)(14,21,17,29)(15,25,28,40)(18,31,24,38)(20,35,32,42)(22,36,33,43)(23,37,26,39)(27,41,34,44);(6,9)(10,12)(13,17,19,14)(15,26,28,23)(16,29,30,21)(18,22,24,33)(20,27,32,34)(25,39,40,37)(31,36,38,43)(35,41,42,44);(5,7,8,11)(6,10,9,12)(16,30)(21,29)(25,40)(31,38)(35,42)(36,43)(37,39)(41,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,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, 7050961);