#10247272

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	#54465	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3846 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10154468);