#9814932

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	#45817	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=h,d^2=i,e^2=g,f^2=h,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, 9722128);