#181204

order	512
nilpotency class	3
isoclinism	isoclinic class of #181129
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	Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1272 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88400);