#9851052

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	#46669	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(17,18)(19,24)(20,25)(21,29)(22,23)(26,31)(27,28)(30,32)(34,37)(35,41)(40,45)(42,47);(1,3,7,4)(2,8,6,9)(5,12,13,11)(10,16,14,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,30)(28,32)(29,31)(33,34,38,37)(35,39,41,43)(36,40,44,45)(42,46,47,48);(18,23)(19,25)(26,29)(27,32)(33,35,38,41)(34,39,37,43)(36,42,44,47)(40,46,45,48);(1,4,7,3)(2,9,6,8)(5,12,13,11)(10,16,14,15)(17,19,22,25)(18,24,23,20)(21,27,31,32)(26,30,29,28)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,47)(45,48);(1,5)(2,10)(3,11)(4,12)(6,14)(7,13)(8,15)(9,16)(17,20,22,24)(18,25,23,19)(21,30,31,28)(26,27,29,32)(33,36)(34,40)(35,42)(37,45)(38,44)(39,46)(41,47)(43,48);(1,6,7,2)(3,9,4,8)(5,10,13,14)(11,15,12,16)(17,21)(18,26)(19,27)(20,28)(22,31)(23,29)(24,30)(25,32)(33,37,38,34)(35,43,41,39)(36,40,44,45)(42,46,47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=gh,e^2=i,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, 9758248);