#10412335

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=h,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10319531);