#10195586

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	#2668	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gi,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10102782);