#10316826

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	#58802	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3868 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gh,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, 10224022);