#9481489

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	#57795	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3767 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)(17,18,23,21)(19,26,24,20)(22,29,31,25)(27,28,32,30)(33,34,38,35)(36,41,44,39)(37,40,45,42)(43,47,48,46);(4,11)(8,14)(10,15)(12,16)(17,19)(18,20)(21,26)(22,27)(23,24)(25,28)(29,30)(31,32)(34,35)(39,41)(40,42)(46,47);(2,7)(3,9)(8,14)(10,15)(17,20,23,26)(18,24,21,19)(22,30,31,28)(25,27,29,32)(33,35,38,34)(36,41,44,39)(37,42,45,40)(43,47,48,46);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,20)(18,19)(21,24)(22,28)(23,26)(25,27)(29,32)(30,31)(33,36,38,44)(34,39,35,41)(37,43,45,48)(40,46,42,47);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,21,23,18)(19,26,24,20)(22,29,31,25)(27,30,32,28);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,22,23,31)(18,25,21,29)(19,27,24,32)(20,28,26,30)(33,37,38,45)(34,40,35,42)(36,43,44,48)(39,46,41,47)

Polycyclic presentation:

2^9|a^2=hi,c^2=g,d^2=i,e^2=h,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, 9388685);