#8097838

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

4132 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=dhi,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8005034);