#6386205

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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	C₂³×C₂²⋊C₄
prime radical	#62815	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293401);