#6465636

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
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3122 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,c^2=ghi,d^2=hi,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, 6372832);