#181228

order	512
nilpotency class	3
isoclinism	isoclinic class of #181131
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1306 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16)(17,18)(19,29,32,24)(20,25)(21,26)(22,28)(23,27)(30,41,34,38)(31,40,33,39)(35,42)(36,43)(37,44)(45,48,46,47);(1,3)(2,6,8,14)(4,11)(5,10)(7,9,12,15)(13,16)(17,19,22,33,23,32,21,31)(18,24,28,40,27,29,26,39)(20,34,35,46,36,30,37,45)(25,41,42,48,43,38,44,47);(3,10)(6,14)(9,15)(11,16)(17,20)(18,25)(19,30)(21,37)(22,35)(23,36)(24,38)(26,44)(27,43)(28,42)(29,41)(31,46)(32,34)(33,45)(39,48)(40,47);(17,21,23,22)(18,26,27,28)(19,31,32,33)(20,35,36,37)(24,39,29,40)(25,42,43,44)(30,45,34,46)(38,47,41,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=i,e^2=hi,g^2=i,b^a=be,e^a=eh,g^a=ghi,c^b=ch,f^b=fi,d^c=di

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, 88424);