#10360948

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

4091 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,16)(13,15)(17,18,23,20)(19,21,24,27)(22,25,31,28)(26,29,30,32)(34,37)(39,43)(40,44)(46,48);(2,7)(3,11)(5,14)(12,15)(17,19,23,24)(18,21,20,27)(22,30,31,26)(25,32,28,29)(34,37)(35,41)(40,44)(42,47);(2,7)(4,9)(8,10)(12,15)(19,24)(21,27)(26,30)(29,32)(33,34,38,37)(35,39,41,43)(36,44,45,40)(42,48,47,46);(1,3,6,11)(2,8,7,10)(4,12,9,15)(5,13,14,16)(17,20,23,18)(19,21,24,27)(22,28,31,25)(26,29,30,32)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,21)(18,24)(19,20)(22,29)(23,27)(25,30)(26,28)(31,32)(33,36,38,45)(34,40,37,44)(35,42,41,47)(39,46,43,48);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,22)(18,25)(19,26)(20,28)(21,29)(23,31)(24,30)(27,32)(33,37,38,34)(35,43,41,39)(36,44,45,40)(42,48,47,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=hi,e^2=h,f^2=gh,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, 10268144);