#181403

order	512
nilpotency class	3
isoclinism	isoclinic class of #181344
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	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1226 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

(1,2,4,8,6,9,5,13)(3,7,12,15,11,16,10,14)(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);(1,3,5,12,6,11,4,10)(2,7,8,14,9,16,13,15)(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);(1,4,6,5)(2,8,9,13)(3,10,11,12)(7,14,16,15)(17,20,22,31)(18,24,25,36)(19,28,29,41)(21,32,33,44)(23,34,27,39)(26,37,38,46)(30,42,43,48)(35,45,40,47);(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=h,f^2=i,g^2=h,b^a=be,e^a=eh,g^a=gi,e^b=ei,f^b=fh,d^c=dhi

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, 88599);