#484414

order	512
nilpotency class	4
isoclinism	isoclinic class of #12093
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
socle	C₂²	C₂×B(4,𝔽₂)
prime radical	#392	C₂³
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

769 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,c^2=g,d^2=i,e^2=g,b^a=be,c^a=cf,e^a=eg,h^a=hi,c^b=ci,e^b=eg,f^b=fhi,e^c=eh,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391610);