#6606064

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574015
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2081 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)(13,14,19,23)(15,20,25,34)(16,28,29,42)(17,22,18,32)(21,30,36,31)(24,37,40,44)(26,35,27,41)(33,38,43,39);(2,4)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,24)(17,26)(18,27)(19,25)(21,33)(22,35)(23,34)(28,37)(29,40)(30,38)(31,39)(32,41)(36,43)(42,44);(6,9)(7,11)(13,16,17,30,19,29,18,31)(14,21,32,42,23,36,22,28)(15,24,26,38,25,40,27,39)(20,33,41,44,34,43,35,37);(5,7)(6,10)(8,11)(9,12)(15,25)(16,29)(20,34)(21,36)(26,27)(28,42)(30,31)(35,41);(6,9)(7,11)(13,17,19,18)(14,22,23,32)(15,26,25,27)(16,30,29,31)(20,35,34,41)(21,28,36,42)(24,38,40,39)(33,37,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,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=dhi,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513260);