#9007646

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3831 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,c^2=g,d^2=g,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8914842);