#8097245

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

4013 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)(3,8)(4,12)(5,9)(6,10)(7,14)(11,13)(15,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3)(2,5)(4,11)(6,9)(7,13)(8,10)(12,15)(14,16)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(35,42)(39,46)(43,47)(44,48);(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,4,6,14)(2,7,10,12)(3,11,9,16)(5,13,8,15)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32);(1,3)(2,8)(4,11)(5,10)(6,9)(7,15)(12,13)(14,16)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,5,6,8)(2,9,10,3)(4,13,14,15)(7,16,12,11)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(33,37)(34,40)(35,44)(36,41)(38,45)(39,47)(42,48)(43,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=g,e^2=h,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, 8004441);