#8097038

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

4096 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,16)(14,15)(17,18,22,26)(19,23,20,24)(21,25,29,32)(27,30,28,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,46)(45,47);(2,7)(3,10)(9,15)(11,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(36,44)(39,46)(43,47)(45,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,13)(12,15)(14,16)(17,19,22,20)(18,23,26,24)(21,27,29,28)(25,30,32,31)(33,35,38,40)(34,37,41,42)(36,43,44,47)(39,45,46,48);(1,3)(2,8)(4,7)(5,11)(6,10)(9,13)(12,15)(14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,36)(34,39)(35,43)(37,45)(38,44)(40,47)(41,46)(42,48);(1,4,6,8)(2,3,7,10)(5,13,14,12)(9,16,15,11)(17,20,22,19)(18,24,26,23)(21,27,29,28)(25,30,32,31)(33,34)(35,37)(36,39)(38,41)(40,42)(43,45)(44,46)(47,48);(1,5,6,14)(2,9,7,15)(3,11,10,16)(4,12,8,13)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,37)(34,40)(35,41)(36,45)(38,42)(39,47)(43,46)(44,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,e^2=g,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, 8004234);