#184017

order	512
nilpotency class	3
isoclinism	isoclinic class of #3055
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×C₈² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

666 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)(20,25)(21,35)(22,27)(23,26)(24,30)(29,42)(31,39)(32,45)(33,41)(34,40)(36,38)(37,43)(44,48)(46,47);(1,3)(2,7)(4,9)(5,11)(6,12)(8,10)(13,16)(14,15)(17,19,22,30,23,31,21,36)(18,24,27,39,26,38,35,28)(20,29,33,43,34,44,32,46)(25,37,41,48,40,47,45,42);(2,8)(3,11)(9,16)(12,15)(17,20,22,33,23,34,21,32)(18,25,27,41,26,40,35,45)(19,29,30,43,31,44,36,46)(24,37,39,48,38,47,28,42);(1,4,6,10,5,13,14,7)(2,3,9,12,8,11,16,15)(18,26)(24,38)(25,40)(27,35)(28,39)(37,47)(41,45)(42,48)

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=ci,d^a=dh,e^a=eh,c^b=ci,e^b=eh

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, 91213);