#9863136

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	#57205	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4104 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)(17,18,23,26)(19,24,22,32)(20,25,30,21)(27,28,29,31)(33,34,39,44)(35,45,47,40)(36,41,37,42)(38,43,48,46);(1,3,7,4)(2,8,9,6)(5,12,14,11)(10,13,16,15)(17,19,23,22)(18,24,26,32)(20,27,30,29)(21,31,25,28)(33,35)(34,40)(36,46)(37,43)(38,42)(39,47)(41,48)(44,45);(2,9)(3,4)(10,16)(11,12)(17,20,23,30)(18,21,26,25)(19,27,22,29)(24,31,32,28)(33,36,39,37)(34,41,44,42)(35,46,47,43)(38,40,48,45);(1,4,7,3)(2,6,9,8)(5,12,14,11)(10,13,16,15)(17,20)(18,25)(19,27)(21,26)(22,29)(23,30)(24,28)(31,32)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,47)(46,48);(1,5,7,14)(2,10,9,16)(3,11,4,12)(6,13,8,15)(17,21,23,25)(18,20,26,30)(19,28,22,31)(24,29,32,27)(33,37,39,36)(34,42,44,41)(35,46,47,43)(38,40,48,45);(1,6)(2,3)(4,9)(5,13)(7,8)(10,11)(12,16)(14,15)(17,22,23,19)(18,24,26,32)(20,29,30,27)(21,31,25,28)(33,38,39,48)(34,43,44,46)(35,42,47,41)(36,40,37,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=i,e^2=ghi,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, 9770332);