#10320403

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

3844 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=hi,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, 10227599);