#6532516

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500002
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

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

2834 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6439712);