#10312370

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	#47390	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3899 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18,22,26)(19,23,27,31)(20,24,30,28)(21,25,32,29)(33,34)(35,44)(36,40)(37,48)(38,43)(39,46)(41,47)(42,45);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19)(18,23)(20,29)(21,24)(22,27)(25,30)(26,31)(28,32)(33,35)(34,39)(36,37)(38,46)(40,42)(41,48)(43,44)(45,47);(2,3)(7,9)(8,10)(14,15)(19,27)(20,30)(23,31)(24,28)(33,36,38,47)(34,40,43,41)(35,37,46,45)(39,42,44,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,24)(19,25)(21,31)(22,30)(23,32)(26,28)(27,29)(33,36)(34,41)(35,45)(37,46)(38,47)(39,42)(40,43)(44,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21)(18,25)(19,28)(20,23)(22,32)(24,27)(26,29)(30,31)(33,34)(35,44)(36,40)(37,48)(38,43)(39,46)(41,47)(42,45);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,18)(19,23)(20,24)(21,29)(22,26)(25,32)(27,31)(28,30)(33,37,38,45)(34,42,43,48)(35,36,46,47)(39,40,44,41)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=gh,d^2=h,e^2=h,f^2=gh,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, 10219566);