#6537054

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535687
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2018	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1569 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:

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

Polycyclic presentation:

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

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, 6444250);