#9481509

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

3903 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)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18,22,23)(19,25,26,20)(21,29,30,24)(27,28,32,31)(33,34)(35,38)(36,37)(39,41)(40,42)(43,46)(44,47)(45,48);(4,11)(8,14)(10,15)(12,16)(17,19,22,26)(18,20,23,25)(21,27,30,32)(24,28,29,31)(33,35,39,42)(34,40,41,38)(36,43,45,47)(37,44,48,46);(2,7)(3,9)(8,14)(10,15)(18,23)(19,26)(24,29)(27,32)(34,41)(36,45)(38,40)(43,47);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,28)(29,30)(31,32)(33,36,39,45)(34,37,41,48)(35,43,42,47)(38,46,40,44);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,20)(18,19)(21,28)(22,25)(23,26)(24,27)(29,32)(30,31)(33,37)(34,36)(35,44)(38,47)(39,48)(40,43)(41,45)(42,46);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,21,22,30)(18,24,23,29)(19,27,26,32)(20,28,25,31)(33,38,39,40)(34,42,41,35)(36,46,45,44)(37,47,48,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388705);