#6575827

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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1991 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)(5,6,8,9)(7,10,11,12)(13,14,17,21)(15,23,27,19)(16,28,30,20)(18,22,31,37)(24,34,25,33)(26,35,42,39)(29,38,32,36)(40,43,41,44);(6,9)(10,12)(15,24)(16,29)(18,31)(19,33)(20,36)(22,37)(23,34)(25,27)(28,38)(30,32)(40,41)(43,44);(2,4)(13,15,17,27)(14,19,21,23)(16,26,30,42)(18,25,31,24)(20,35,28,39)(22,34,37,33)(29,40,32,41)(36,43,38,44);(5,7,8,11)(6,10,9,12)(15,25)(16,29)(18,31)(19,34)(20,36)(22,37)(23,33)(24,27)(26,42)(28,38)(30,32)(35,39);(13,16,17,30)(14,20,21,28)(15,26,27,42)(18,32,31,29)(19,35,23,39)(22,38,37,36)(24,40,25,41)(33,43,34,44)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6483023);