#361621

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#135	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gi,d^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,g^c=gi,h^c=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, 268817);