#9947441

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	#59599	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=i,e^2=h,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, 9854637);