#7143001

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#45248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1981 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:

(2,4)(7,11)(10,12)(13,14)(15,20)(16,26)(17,22)(18,31)(19,23)(21,24)(25,34)(27,40)(28,33)(29,37)(30,38)(32,36)(35,39)(41,44)(42,43);(1,2)(3,4)(6,9)(7,11)(13,15,19,18)(14,20,23,31)(16,24,28,32)(17,29,30,25)(21,33,36,26)(22,37,38,34)(27,41,42,39)(35,40,44,43);(5,6)(7,10)(8,9)(11,12)(13,16,19,28)(14,21,23,36)(15,24,18,32)(17,27,30,42)(20,33,31,26)(22,35,38,44)(25,39,29,41)(34,43,37,40);(2,4)(6,9)(10,12)(13,17,19,30)(14,22,23,38)(15,25,18,29)(16,27,28,42)(20,34,31,37)(21,35,36,44)(24,39,32,41)(26,40,33,43);(5,7,8,11)(6,10,9,12)(14,23)(17,30)(20,31)(21,36)(25,29)(26,33)(27,42)(39,41)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,g^c=gi,e^d=ei

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, 7050197);