#182040

order	512
nilpotency class	3
isoclinism	isoclinic class of #17386
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

984 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)(3,6)(4,8)(5,7)(9,13)(10,12)(11,14)(15,16)(17,18,21,25,22,26,33,39)(19,27,30,41,29,23,34,36)(20,24,32,38,31,37,44,47)(28,40,43,48,42,35,45,46);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,19)(18,23,26,27)(20,28)(21,30)(22,29)(24,35,37,40)(25,36,39,41)(31,42)(32,43)(33,34)(38,46,47,48)(44,45);(2,7)(6,12)(8,14)(13,16)(17,20,21,32,22,31,33,44)(18,24,25,38,26,37,39,47)(19,28,30,43,29,42,34,45)(23,35,36,46,27,40,41,48);(20,31)(24,37)(28,42)(32,44)(35,40)(38,47)(43,45)(46,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,g^2=i,b^a=be,c^a=ci,g^a=gh,e^b=eh,d^c=dh

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, 89236);