#189541

order	512
nilpotency class	3
isoclinism	isoclinic class of #189531
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1180 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)(3,7,8,5)(4,6)(9,10,14,18,15,19,13,28)(11,20,23,30,24,34,22,16)(12,17,27,32,25,31,26,38)(21,33,37,39,35,40,36,29);(9,11,14,23,15,24,13,22)(10,16,28,34,19,30,18,20)(12,21,27,37,25,35,26,36)(17,29,38,40,31,39,32,33);(1,3)(2,5)(4,8)(6,7)(9,12)(10,17)(11,21)(13,26)(14,27)(15,25)(16,29)(18,32)(19,31)(20,33)(22,36)(23,37)(24,35)(28,38)(30,39)(34,40);(12,25)(17,31)(21,35)(26,27)(29,39)(32,38)(33,40)(36,37)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 96737);