#6576870

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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	#36078	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2720 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=i,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=ehi,h^d=hi,h^e=hi

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, 6484066);