#232518

order	512
nilpotency class	3
isoclinism	isoclinic class of #232515
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	C₂³×C₄	C₂⁴
prime radical	#60359	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2209 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139714);