#8097204

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)(3,11)(4,10)(5,9)(6,12)(7,13)(8,14)(15,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,31)(30,32)(33,34,38,41)(35,39,42,46)(36,37,40,45)(43,44,47,48);(1,3,8,5)(2,9,14,11)(4,13,15,6)(7,10,12,16)(17,19,24,22)(18,25,21,20)(23,27,31,30)(26,32,29,28)(35,42)(39,46)(43,47)(44,48);(1,4,8,15)(2,10,14,16)(3,13,5,6)(7,9,12,11)(17,20)(18,22)(19,21)(23,28)(24,25)(26,30)(27,29)(31,32)(33,35)(34,39)(36,43)(37,44)(38,42)(40,47)(41,46)(45,48);(1,5,8,3)(2,11,14,9)(4,6,15,13)(7,16,12,10)(17,21,24,18)(19,25,22,20)(23,29,31,26)(27,32,30,28);(1,6)(2,12)(3,4)(5,15)(7,14)(8,13)(9,16)(10,11)(17,22,24,19)(18,25,21,20)(23,27,31,30)(26,28,29,32)(33,36)(34,37)(35,43)(38,40)(39,44)(41,45)(42,47)(46,48);(1,7,8,12)(2,13,14,6)(3,10,5,16)(4,9,15,11)(17,23,24,31)(18,26,21,29)(19,27,22,30)(20,28,25,32)(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=g,c^2=gh,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, 8004400);