#9851273

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	#49733	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3950 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=i,e^2=ghi,f^2=g,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, 9758469);