#10320366

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3858 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,25)(19,22,20,24)(21,26,30,32)(27,29,28,31)(33,34)(35,44)(36,40)(37,48)(38,43)(39,46)(41,47)(42,45);(3,9)(7,13)(10,15)(14,16)(17,19,23,20)(18,24,25,22)(21,28,30,27)(26,29,32,31)(33,35)(34,39)(36,37)(38,46)(40,42)(41,48)(43,44)(45,47);(2,6)(7,13)(8,11)(14,16)(18,25)(19,20)(26,32)(27,28)(33,36,38,47)(34,40,43,41)(35,37,46,45)(39,42,44,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,19)(18,22,25,24)(21,28,30,27)(26,29,32,31)(33,36)(34,41)(35,45)(37,46)(38,47)(39,42)(40,43)(44,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21)(18,26)(19,27)(20,28)(22,29)(23,30)(24,31)(25,32)(33,34)(35,44)(36,40)(37,48)(38,43)(39,46)(41,47)(42,45);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,22)(18,19)(20,25)(21,29)(23,24)(26,27)(28,32)(30,31)(33,37,38,45)(34,42,43,48)(35,36,46,47)(39,40,44,41)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=gi,d^2=i,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, 10227562);