#8700331

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3880 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=gh,e^2=h,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8607527);