#232928

order	512
nilpotency class	3
isoclinism	isoclinic class of #232927
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#60940	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1577 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)(3,5)(4,7)(6,8)(9,10)(11,20)(12,17)(13,28)(14,19)(15,18)(16,22)(21,37)(23,31)(24,36)(25,30)(26,35)(27,33)(29,34)(32,38)(39,40);(9,11,15,24)(10,16,18,31)(12,25,27,39)(13,23,14,22)(17,32,33,37)(19,36,28,20)(21,29,38,26)(30,34,40,35);(2,5)(3,8)(6,7)(9,12)(10,17)(11,21)(13,26)(14,29)(15,27)(16,30)(18,33)(19,34)(20,37)(22,25)(23,39)(24,38)(28,35)(31,40)(32,36);(10,18)(16,31)(17,33)(19,28)(20,36)(30,40)(32,37)(34,35)

Polycyclic presentation:

2^9|b^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fi,c^b=cg,e^b=eh,g^b=gh,f^c=fi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140124);