#181386

order	512
nilpotency class	3
isoclinism	isoclinic class of #181344
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#95
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1332 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,29,32,24)(20,25)(21,26)(22,28)(23,27)(30,43,45,37)(31,39,33,38)(34,40)(35,41)(36,42)(44,48,46,47);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,19,22,33,23,32,21,31)(18,24,28,39,27,29,26,38)(20,30,34,44,35,45,36,46)(25,37,40,47,41,43,42,48);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,37)(26,42)(27,41)(28,40)(29,43)(31,46)(32,45)(33,44)(38,48)(39,47);(17,21,23,22)(18,26,27,28)(19,31,32,33)(20,34,35,36)(24,38,29,39)(25,40,41,42)(30,44,45,46)(37,47,43,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,g^2=h,b^a=be,e^a=eh,g^a=gi,e^b=ei,f^b=fh,d^c=dh

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, 88582);