#9794350

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4085 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=ghi,c^2=h,d^2=gh,e^2=h,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, 9701546);