#9685534

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	#44172	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3988 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,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, 9592730);