#8097101

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

4169 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)(19,26)(20,23)(21,24)(22,25)(27,30)(28,31)(29,32)(33,34,39,38)(35,43,46,40)(36,41,45,47)(37,48,44,42);(4,12)(8,15)(11,13)(14,16)(18,20)(19,24)(25,28)(27,32)(33,35)(34,40)(36,44)(37,45)(38,43)(39,46)(41,48)(42,47);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,36,39,45)(34,41,38,47)(35,44,46,37)(40,48,43,42);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,30,32,29)(33,37,39,44)(34,42,38,48)(35,45,46,36)(40,47,43,41);(1,5,6,3)(2,7,9,10)(4,13,12,11)(8,14,15,16)(17,21,23,26)(18,24,20,19)(22,30,31,29)(25,27,28,32)(33,35)(34,43)(36,44)(37,45)(38,40)(39,46)(41,42)(47,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,22,23,31)(18,25,20,28)(19,27,24,32)(21,29,26,30)(33,38,39,34)(35,43,46,40)(36,47,45,41)(37,48,44,42)

Polycyclic presentation:

2^9|a^2=gi,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, 8004297);