#9847671

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

3939 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=h,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,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, 9754867);