#6585504

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	#36015	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1848 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)(5,6)(7,11)(8,10)(9,12)(13,14,17,21)(15,23,27,19)(16,20,29,36)(18,22,31,38)(24,34,26,32)(25,39,41,33)(28,35,30,37)(40,44,42,43);(5,7,9,8)(6,10,12,11)(13,15,17,27)(14,19,21,23)(16,25,29,41)(18,24,31,26)(20,33,36,39)(22,32,38,34)(28,40,30,42)(35,43,37,44);(1,2,3,4)(15,24)(16,28)(18,31)(19,32)(20,35)(22,38)(23,34)(26,27)(29,30)(36,37)(40,42)(43,44);(5,7,9,8)(6,11,12,10)(13,16)(14,20)(15,25)(17,29)(18,28)(19,33)(21,36)(22,35)(23,39)(24,40)(26,42)(27,41)(30,31)(32,43)(34,44)(37,38);(5,8,9,7)(6,10,12,11)(15,26)(16,28)(18,31)(19,34)(20,35)(22,38)(23,32)(24,27)(25,41)(29,30)(33,39)(36,37)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,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, 6492700);