#9947394

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	#48161	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3967 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,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, 9854590);