#10368638

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

4233 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:

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

Polycyclic presentation:

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

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, 10275834);