#8097310

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

4198 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,d^2=gh,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, 8004506);