#483676

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	#3106	C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ #2162 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=f,e^2=g,f^2=h,h^2=i,b^a=be,e^a=eg,d^b=di,e^b=eg,d^c=dg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320272, #72322030, #72322049
maximal normal subgroup of	#72320272, #72322030, #72322049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390872);