#9815144

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	#45833	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4044 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=g,d^2=ghi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eghi,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, 9722340);