#392375

order	512
nilpotency class	3
isoclinism	isoclinic class of #188764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#381
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1334 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)(11,19,22,15)(12,23,24,38)(13,18)(14,17)(16,25,29,39)(20,33)(21,32,36,28)(26,30)(27,35)(31,40)(34,37);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,27)(17,19)(18,28)(23,37)(24,34)(25,35)(26,36)(29,31)(30,32)(33,38)(39,40);(2,6)(5,8)(12,24)(16,29)(20,34)(23,38)(25,39)(27,31)(33,37)(35,40);(9,12)(10,16,17,29)(11,20)(13,25)(14,24)(15,27,19,31)(18,23,30,38)(21,35)(22,34)(26,39)(28,37,32,33)(36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=e,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,d^c=di,h^d=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, 299571);