#9720177

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

4130 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,24,23)(19,28,22,25)(20,26,21,27)(29,32,30,31)(35,42)(37,39)(43,48)(45,47);(3,9)(7,13)(10,15)(14,16)(17,19,24,22)(18,25,23,28)(20,29,21,30)(26,31,27,32)(33,34)(35,37)(36,44)(38,41)(39,42)(40,46)(43,45)(47,48);(2,6)(7,13)(8,11)(14,16)(17,20,24,21)(18,26,23,27)(19,29,22,30)(25,31,28,32)(33,35,38,42)(34,39,41,37)(36,43,46,48)(40,45,44,47);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,21,24,20)(18,27,23,26)(19,30,22,29)(25,32,28,31)(33,35)(34,37)(36,43)(38,42)(39,41)(40,47)(44,45)(46,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,22,24,19)(18,25,23,28)(20,30,21,29)(26,31,27,32)(33,36,38,46)(34,40,41,44)(35,43,42,48)(37,47,39,45);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,23,24,18)(19,28,22,25)(20,27,21,26)(29,32,30,31)(33,37)(34,35)(36,45)(38,39)(40,48)(41,42)(43,44)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gi,d^2=g,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,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, 9627373);