#184022

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₄²×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

658 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 91218);