#10360925

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

4163 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)(3,9)(4,8)(5,6)(7,10)(11,16)(12,14)(13,15)(17,18,23,20)(19,21,24,27)(22,25,31,28)(26,29,30,32)(34,39)(35,40)(37,45)(44,48);(2,6)(3,10)(8,14)(11,13)(17,19,23,24)(18,21,20,27)(22,30,31,26)(25,32,28,29)(33,34)(35,42)(36,40)(37,41)(38,39)(43,48)(44,47)(45,46);(4,12)(8,14)(11,13)(15,16)(19,24)(21,27)(26,30)(29,32)(33,35,38,40)(34,36,39,42)(37,43,45,47)(41,44,46,48);(1,3)(2,7)(4,13)(5,10)(6,9)(8,16)(11,12)(14,15)(17,20,23,18)(19,21,24,27)(22,28,31,25)(26,29,30,32)(33,34)(35,36)(37,41)(38,39)(40,42)(43,44)(45,46)(47,48);(1,4,5,12)(2,8,6,14)(3,11,10,13)(7,15,9,16)(17,21)(18,24)(19,20)(22,29)(23,27)(25,30)(26,28)(31,32)(33,36)(34,40)(35,39)(37,44)(38,42)(41,47)(43,46)(45,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,22)(18,25)(19,26)(20,28)(21,29)(23,31)(24,30)(27,32)(33,37,38,45)(34,41,39,46)(35,43,40,47)(36,44,42,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=h,d^2=hi,e^2=gh,f^2=h,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, 10268121);