#6399081

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61234	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=efi,g^a=gi,c^b=ci,d^b=df,e^b=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6306277);