#8733936

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

4152 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)(3,7,11,8)(4,5,9,15)(12,13,16,14)(17,18)(19,24)(20,23)(21,25)(22,27)(26,31)(28,29)(30,32)(33,34,38,37)(35,40,43,39)(36,44,45,41)(42,46,48,47);(1,3)(2,7)(4,12)(5,13)(6,11)(8,10)(9,16)(14,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,32)(29,31)(33,35,38,43)(34,39,37,40)(36,42,45,48)(41,46,44,47);(3,11)(7,8)(12,16)(13,14)(20,28)(23,29)(26,32)(30,31)(33,35,38,43)(34,39,37,40)(36,42,45,48)(41,46,44,47);(1,3)(2,8)(4,12)(5,14)(6,11)(7,10)(9,16)(13,15)(17,20,21,28)(18,23,25,29)(19,26,27,32)(22,30,24,31)(33,35)(34,40)(36,42)(37,39)(38,43)(41,47)(44,46)(45,48);(1,4)(2,5)(3,12)(6,9)(7,13)(8,14)(10,15)(11,16)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,32)(29,30)(33,36,38,45)(34,41,37,44)(35,42,43,48)(39,46,40,47);(1,5)(2,9)(3,13)(4,10)(6,15)(7,16)(8,12)(11,14)(17,18)(19,22)(20,29)(21,25)(23,28)(24,27)(26,31)(30,32)(33,37,38,34)(35,40,43,39)(36,41,45,44)(42,46,48,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

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, 8641132);