#6585515

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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1826 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)(8,12)(9,10)(13,14,17,20)(15,23,26,19)(16,21,30,35)(18,22,31,36)(24,32,27,34)(25,39,41,33)(28,38,29,37)(40,43,42,44);(6,10)(11,12)(13,15,17,26)(14,19,20,23)(16,25,30,41)(18,24,31,27)(21,33,35,39)(22,34,36,32)(28,42,29,40)(37,44,38,43);(1,2,3,4)(5,7,9,8)(6,11,10,12)(14,20)(15,24)(16,28)(18,31)(19,32)(21,37)(23,34)(25,41)(26,27)(29,30)(35,38)(43,44);(5,7,9,8)(6,11,10,12)(13,16)(14,21)(15,25)(17,30)(18,29)(19,33)(20,35)(22,37)(23,39)(24,40)(26,41)(27,42)(28,31)(32,43)(34,44)(36,38);(5,8,9,7)(6,12,10,11)(15,24)(16,29)(18,31)(19,34)(21,37)(22,36)(23,32)(26,27)(28,30)(35,38)(40,42)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6492711);