#6348016

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347957
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#62831	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2679 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=hi,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,f^a=fh,d^b=dh,f^b=fh,g^b=gi,e^c=eh,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255212);