#8701230

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

3882 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)(3,10)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,18,22,26)(19,24,20,23)(21,25,30,29)(27,32,28,31);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,34)(35,41)(36,37)(38,42)(39,45)(40,46)(43,47)(44,48);(17,19,22,20)(18,23,26,24)(21,27,30,28)(25,31,29,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,47)(41,42)(46,48);(1,4,7,2)(3,8,5,10)(6,9,14,12)(11,16,13,15)(17,20,22,19)(18,24,26,23)(21,27,30,28)(25,31,29,32)(33,36,38,46)(34,40,42,37)(35,43,45,48)(39,47,41,44);(1,5,7,3)(2,8,4,10)(6,11,14,13)(9,16,12,15)(17,18)(19,23)(20,24)(21,29)(22,26)(25,30)(27,32)(28,31)(33,35)(34,41)(36,43)(37,47)(38,45)(39,42)(40,44)(46,48);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,21,22,30)(18,25,26,29)(19,27,20,28)(23,31,24,32)(33,37,38,40)(34,36,42,46)(35,44,45,47)(39,43,41,48)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=gi,e^2=gh,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, 8608426);