#6465633

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438943
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

3234 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)(9,10,14,20,15,21,29,19)(11,22,26,34,27,38,25,16)(12,17,13,18,30,37,31,36)(23,35,28,40,39,33,24,32);(2,4)(9,11)(10,16)(12,28)(13,23)(14,25)(15,27)(17,35)(18,32)(19,22)(20,38)(21,34)(24,30)(26,29)(31,39)(33,37)(36,40);(5,6,7,8)(9,12,29,31,15,30,14,13)(10,17,19,36,21,37,20,18)(11,23,25,24,27,39,26,28)(16,32,38,33,34,40,22,35);(9,13,14,30,15,31,29,12)(10,18,20,37,21,36,19,17)(11,24,26,23,27,28,25,39)(16,33,22,32,34,35,38,40);(9,14,15,29)(10,19,21,20)(11,25,27,26)(12,13,30,31)(16,22,34,38)(17,36,37,18)(23,24,39,28)(32,35,40,33)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6372829);