#483672

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
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

440 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)(19,24,29,36,30,37,42,46,28,31,38,43,41,44,47,48);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,24)(20,31)(21,29)(22,30)(23,28)(25,38)(26,36)(27,37)(32,43)(33,44)(34,42)(35,41)(39,47)(40,46)(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)(19,24,29,36,30,37,42,46,28,31,38,43,41,44,47,48);(3,10)(7,14)(11,16)(13,15)(18,20)(19,28)(26,32)(27,33)(29,38)(30,41)(40,45)(42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287023, #72322022, #72322049
maximal normal subgroup of	#72287023, #72322022, #72322049

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390868);