#10320402

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

3837 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)(3,11)(4,14)(5,8)(6,10)(7,9)(12,15)(13,16)(17,18,23,21)(19,26,25,22)(20,28,31,24)(27,30,29,32)(33,34,39,37)(35,40,43,36)(38,45,47,41)(42,44,46,48);(1,3,8,7)(2,9,5,11)(4,13,6,12)(10,16,14,15)(17,19,23,25)(18,22,21,26)(20,29,31,27)(24,32,28,30)(33,35,39,43)(34,40,37,36)(38,42,47,46)(41,48,45,44);(1,4,8,6)(2,10,5,14)(3,12,7,13)(9,15,11,16)(17,18,23,21)(19,22,25,26)(20,28,31,24)(27,32,29,30)(33,36,39,40)(34,35,37,43)(38,44,47,48)(41,42,45,46);(1,5,8,2)(3,11,7,9)(4,10,6,14)(12,15,13,16)(17,20,23,31)(18,24,21,28)(19,27,25,29)(22,30,26,32)(33,37,39,34)(35,40,43,36)(38,45,47,41)(42,48,46,44);(1,6,8,4)(2,10,5,14)(3,13,7,12)(9,15,11,16)(17,21,23,18)(19,22,25,26)(20,24,31,28)(27,32,29,30)(33,35)(34,40)(36,37)(38,46)(39,43)(41,44)(42,47)(45,48);(1,7,8,3)(2,11,5,9)(4,13,6,12)(10,16,14,15)(17,22)(18,25)(19,21)(20,30)(23,26)(24,29)(27,28)(31,32)(33,38)(34,41)(35,42)(36,44)(37,45)(39,47)(40,48)(43,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=ghi,d^2=ghi,e^2=g,f^2=h,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, 10227598);