#6465663

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

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 ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3184 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=ghi,d^2=h,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372859);