#9815386

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
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3874 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^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, 9722582);