#10247404

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

3782 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:

(1,2,7,11)(3,12,5,8)(4,13,15,9)(6,10,16,14)(17,18,23,20)(19,24,26,22)(21,25,31,28)(27,30,29,32);(1,3,7,5)(2,8,11,12)(4,14,15,10)(6,13,16,9)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,32,28,30)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,45,47);(18,20)(19,26)(25,28)(27,29)(33,35)(34,39)(36,41)(37,42)(38,43)(40,47)(44,48)(45,46);(1,4,7,15)(2,9,11,13)(3,10,5,14)(6,8,16,12)(17,20,23,18)(19,24,26,22)(21,25,31,28)(27,32,29,30)(33,36,38,39)(34,35,41,43)(37,44,46,47)(40,42,48,45);(1,5,7,3)(2,8,11,12)(4,10,15,14)(6,13,16,9)(17,21)(18,25)(19,27)(20,28)(22,32)(23,31)(24,30)(26,29)(33,35)(34,36)(37,45)(38,43)(39,41)(40,47)(42,46)(44,48);(1,6)(2,10)(3,13)(4,8)(5,9)(7,16)(11,14)(12,15)(17,22,23,24)(18,26,20,19)(21,30,31,32)(25,27,28,29)(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,d^2=gi,e^2=g,f^2=h,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, 10154600);