#9815131

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	#43378	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=ghi,e^2=g,f^2=gh,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, 9722327);