#181203

order	512
nilpotency class	3
isoclinism	isoclinic class of #181127
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	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

1256 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)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,21,26,23,28,22,27)(19,29,31,24,33,38,32,39)(20,25,35,41,36,42,34,40)(30,43,45,37,46,47,44,48);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(17,19,22,32,23,33,21,31)(18,24,26,38,28,39,27,29)(20,30,34,44,36,46,35,45)(25,37,41,47,42,48,40,43);(17,20,23,36)(18,25,28,42)(19,30,33,46)(21,35,22,34)(24,37,39,48)(26,41,27,40)(29,43,38,47)(31,45,32,44);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,34,36,35)(24,38,39,29)(25,40,42,41)(30,44,46,45)(37,43,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88399);