#9794267

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	#54516	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(3,10)(4,7)(5,9)(6,8)(11,16)(12,13)(14,15)(17,18,24,21)(19,22,25,20)(23,29,31,26)(27,28,32,30)(34,40)(36,44)(41,46)(42,45);(1,3,7,6)(2,8,4,10)(5,11,13,14)(9,15,12,16)(17,19)(18,22)(20,21)(23,27)(24,25)(26,30)(28,29)(31,32)(33,34)(35,41)(36,38)(37,46)(39,40)(42,43)(44,48)(45,47);(2,4)(8,10)(9,12)(15,16)(17,20,24,22)(18,25,21,19)(23,28,31,30)(26,32,29,27)(33,35,39,37)(34,41,40,46)(36,42,44,45)(38,43,48,47);(1,4,7,2)(3,10,6,8)(5,12,13,9)(11,16,14,15)(17,21,24,18)(19,20,25,22)(23,29,31,26)(27,28,32,30)(33,36)(34,38)(35,42)(37,45)(39,44)(40,48)(41,43)(46,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,22,24,20)(18,25,21,19)(23,28,31,30)(26,27,29,32)(33,37,39,35)(34,41,40,46)(36,45,44,42)(38,43,48,47);(1,6,7,3)(2,10,4,8)(5,11,13,14)(9,15,12,16)(17,23,24,31)(18,26,21,29)(19,27,25,32)(20,28,22,30)(33,38)(34,36)(35,43)(37,47)(39,48)(40,44)(41,42)(45,46)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=h,e^2=gh,f^2=h,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, 9701463);