#9685588

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	#54002	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3948 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)(3,5)(4,7)(6,12)(8,10)(9,14)(11,15)(13,16)(18,20)(24,25)(26,28)(31,32)(33,34,39,40)(35,44,38,41)(36,46,47,42)(37,43,48,45);(1,3)(2,8)(4,5)(6,11)(7,10)(9,15)(12,13)(14,16)(17,18,23,20)(19,24,21,25)(22,26,30,28)(27,31,29,32)(34,40)(37,48)(41,44)(42,46);(2,4)(3,10)(9,12)(11,16)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,39,38)(34,41,40,44)(36,43,47,45)(37,42,48,46);(1,4,7,2)(3,5,10,8)(6,12,14,9)(11,13,16,15)(17,20,23,18)(19,25,21,24)(22,28,30,26)(27,32,29,31)(33,36,39,47)(34,42,40,46)(35,45,38,43)(37,44,48,41);(1,5)(2,3)(4,10)(6,13)(7,8)(9,11)(12,16)(14,15)(17,21,23,19)(18,25,20,24)(22,27,30,29)(26,31,28,32)(33,37)(34,43)(35,42)(36,44)(38,46)(39,48)(40,45)(41,47);(1,6,7,14)(2,9,4,12)(3,11,10,16)(5,13,8,15)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,38,39,35)(34,44,40,41)(36,43,47,45)(37,42,48,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gi,d^2=g,e^2=gh,f^2=g,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, 9592784);