#9711985

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

3939 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=hi,d^2=gh,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=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, 9619181);