#9727144

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
prime radical	#54419	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3993 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)(17,18,22,25)(19,26,23,20)(21,24,30,32)(27,31,29,28)(35,44)(37,47)(39,40)(42,48);(3,9)(7,13)(10,15)(14,16)(17,19)(18,20)(21,29)(22,23)(24,31)(25,26)(27,30)(28,32)(33,34,38,43)(35,40,44,39)(36,45,46,41)(37,42,47,48);(2,6)(7,13)(8,11)(14,16)(17,20,22,26)(18,23,25,19)(21,28,30,31)(24,29,32,27)(33,35,38,44)(34,39,43,40)(36,42,46,48)(37,41,47,45);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,19)(21,28)(22,26)(23,25)(24,27)(29,32)(30,31)(33,35)(34,40)(36,42)(37,41)(38,44)(39,43)(45,47)(46,48);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21)(18,24)(19,27)(20,28)(22,30)(23,29)(25,32)(26,31)(33,36,38,46)(34,41,43,45)(35,42,44,48)(37,39,47,40);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,19)(18,20)(21,29)(22,23)(24,31)(25,26)(27,30)(28,32)(33,37)(34,42)(35,41)(36,40)(38,47)(39,46)(43,48)(44,45)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gi,e^2=i,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, 9634340);