#9706360

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	#40705	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3937 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=gh,d^2=gh,e^2=ghi,f^2=gh,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, 9613556);