#10247389

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	#52332	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3995 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=gi,e^2=g,f^2=gh,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, 10154585);