#6584045

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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	C₂×#3050	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2110 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6491241);