#10360927

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

4239 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,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, 10268123);