#9685532

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
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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,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, 9592728);