#8097102

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4161 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)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18,23,21)(19,26,24,20)(22,25,30,29)(27,32,31,28)(33,34)(35,44)(36,41)(37,47)(38,42)(39,43)(40,46)(45,48);(4,12)(8,15)(11,13)(14,16)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,32)(30,31)(33,35)(34,37)(36,45)(38,46)(39,47)(40,48)(41,42)(43,44);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,18,23,21)(19,20,24,26)(22,25,30,29)(27,28,31,32)(33,36)(34,40)(35,38)(37,42)(39,46)(41,43)(44,48)(45,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20,23,26)(18,24,21,19)(22,28,30,32)(25,31,29,27)(33,37,39,44)(34,35,43,47)(36,42,46,48)(38,41,45,40);(1,5,6,3)(2,7,9,10)(4,13,12,11)(8,14,15,16)(17,21,23,18)(19,26,24,20)(22,25,30,29)(27,28,31,32)(33,36)(34,41)(35,45)(37,42)(38,47)(39,46)(40,43)(44,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,22,23,30)(18,25,21,29)(19,27,24,31)(20,28,26,32)(33,38,39,45)(34,42,43,48)(35,46,47,36)(37,41,44,40)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8004298);