#10320348

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

3835 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gi,d^2=hi,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,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, 10227544);