#9815117

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

4127 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9722313);