#9481511

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	#57573	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3907 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^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, 9388707);