#181385

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

1346 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)(19,27,29,23)(20,24)(21,26)(22,25)(28,39,41,34)(30,40,43,35)(31,36)(32,37)(33,38)(42,47,48,45)(44,46);(1,3)(2,6,8,9)(4,10)(5,11)(7,13,14,15)(12,16)(17,19,21,30,22,29,33,43)(18,23,26,35,25,27,38,40)(20,28,32,42,31,41,44,48)(24,34,37,45,36,39,46,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,24)(19,28)(21,32)(22,31)(23,34)(25,36)(26,37)(27,39)(29,41)(30,42)(33,44)(35,45)(38,46)(40,47)(43,48);(20,31)(24,36)(28,41)(32,44)(34,39)(37,46)(42,48)(45,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,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, 88581);