#8741947

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

4120 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=g,e^2=h,f^2=g,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, 8649143);