#7003557

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	#45236	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1883 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)(13,14)(15,25)(16,21)(17,22)(18,33)(19,24)(20,30)(23,26)(27,40)(28,38)(29,42)(31,36)(32,41)(34,35)(37,43)(39,44);(5,6)(7,11)(8,10)(9,12)(13,15,26,43,19,29,18,30)(14,20,33,42,24,37,23,25)(16,27,44,32,28,17,31,34)(21,35,36,22,38,41,39,40);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,19,28)(14,21,24,38)(15,27,29,17)(18,31,26,44)(20,35,37,41)(22,25,40,42)(23,39,33,36)(30,34,43,32);(7,12)(9,11)(13,17,19,27)(14,22,24,40)(15,28,29,16)(18,34,26,32)(20,36,37,39)(21,25,38,42)(23,41,33,35)(30,31,43,44);(6,10)(9,11)(15,29)(16,28)(20,37)(21,38)(25,42)(30,43)(31,44)(36,39)

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^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=ehi,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, 6910753);