#9726255

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	#2669	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(3,4)(5,9)(6,8)(7,11)(10,12)(13,15)(14,16)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,31)(29,32)(35,37)(39,41)(43,44)(46,47);(1,3)(2,7)(4,8)(5,13)(6,11)(9,12)(10,14)(15,16)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,32,28,30)(33,34,38,42)(35,39,37,41)(36,40,45,48)(43,46,44,47);(2,8)(4,7)(9,16)(12,15)(18,20)(19,26)(25,28)(27,29)(33,35,38,37)(34,39,42,41)(36,44,45,43)(40,47,48,46);(1,4,6,7)(2,3,8,11)(5,12,14,15)(9,10,16,13)(17,20,23,18)(19,22,26,24)(21,28,31,25)(27,32,29,30)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(1,5)(2,9)(3,10)(4,12)(6,14)(7,15)(8,16)(11,13)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,32,24,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3)(2,7)(4,8)(5,13)(6,11)(9,12)(10,14)(15,16)(17,22)(18,19)(20,26)(21,30)(23,24)(25,29)(27,28)(31,32)(33,37,38,35)(34,41,42,39)(36,44,45,43)(40,47,48,46)

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=h,e^2=i,f^2=ghi,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, 9633451);