#375308

order	512
nilpotency class	3
isoclinism	isoclinic class of #24802
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

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	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=egi,c^2=fh,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^c=fi,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, 282504);