#9794344

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

3953 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)(17,18)(19,20)(21,24)(22,30)(23,26)(25,31)(27,32)(28,29)(33,34,38,42)(35,39,37,41)(36,40,45,44)(43,47,46,48);(1,2,7,4)(3,8,5,10)(6,9,14,12)(11,15,13,16)(17,18)(19,20)(21,24)(22,25)(23,26)(27,28)(29,32)(30,31)(33,34)(35,39)(36,44)(37,41)(38,42)(40,45)(43,48)(46,47);(1,3,7,5)(2,8,4,10)(6,11,14,13)(9,15,12,16)(17,19,23,21)(18,24,26,20)(22,27,31,29)(25,32,30,28)(33,35,38,37)(34,39,42,41)(36,43,45,46)(40,47,44,48);(1,4,7,2)(3,8,5,10)(6,12,14,9)(11,15,13,16)(17,20,23,24)(18,19,26,21)(22,28,31,32)(25,27,30,29)(33,34)(35,39)(36,40)(37,41)(38,42)(43,47)(44,45)(46,48);(1,5,7,3)(2,8,4,10)(6,13,14,11)(9,15,12,16)(17,21,23,19)(18,24,26,20)(22,27,31,29)(25,28,30,32)(33,36)(34,40)(35,43)(37,46)(38,45)(39,47)(41,48)(42,44);(1,6,7,14)(2,9,4,12)(3,11,5,13)(8,15,10,16)(17,22,23,31)(18,25,26,30)(19,27,21,29)(20,28,24,32)(33,37,38,35)(34,41,42,39)(36,43,45,46)(40,47,44,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=h,e^2=h,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, 9701540);