#6465646

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3100 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)(9,10,14,18,15,19,28,13)(11,20,22,31,23,35,29,16)(12,17,27,34,26,33,36,25)(21,32,24,38,37,40,39,30);(2,4)(9,11,15,23)(10,16,19,31)(12,24,26,39)(13,20,18,35)(14,29,28,22)(17,32,33,40)(21,36,37,27)(25,38,34,30);(5,6,7,8)(9,12,14,27,15,26,28,36)(10,17,18,34,19,33,13,25)(11,21,22,24,23,37,29,39)(16,30,20,32,31,38,35,40);(9,13,28,19,15,18,14,10)(11,16,29,35,23,31,22,20)(12,25,36,33,26,34,27,17)(21,30,39,40,37,38,24,32);(12,26)(17,33)(21,37)(24,39)(25,34)(27,36)(30,38)(32,40)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gh,d^2=hi,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

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, 6372842);