#9851001

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

4041 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)(3,10)(4,13)(5,8)(6,14)(7,9)(11,12)(15,16)(17,18,23,20)(19,21,24,26)(22,28,31,25)(27,30,32,29)(33,34,38,42)(35,43,45,39)(36,46,47,40)(37,41,44,48);(1,3,7,5)(2,8,9,10)(4,11,14,15)(6,12,13,16)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,30)(31,32)(33,35,38,45)(34,39,42,43)(36,37,47,44)(40,41,46,48);(2,9)(4,14)(8,10)(11,15)(18,20)(19,24)(25,28)(27,32)(33,36,38,47)(34,40,42,46)(35,44,45,37)(39,48,43,41);(1,4)(2,6)(3,11)(5,15)(7,14)(8,12)(9,13)(10,16)(17,20,23,18)(19,26,24,21)(22,28,31,25)(27,30,32,29)(33,36)(34,40)(35,37)(38,47)(39,41)(42,46)(43,48)(44,45);(1,5,7,3)(2,8,9,10)(4,15,14,11)(6,12,13,16)(17,21,23,26)(18,24,20,19)(22,30,31,29)(25,27,28,32)(33,37)(34,41)(35,36)(38,44)(39,40)(42,48)(43,46)(45,47);(1,6)(2,4)(3,12)(5,16)(7,13)(8,11)(9,14)(10,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,31)(24,32)(26,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,48)(46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=gh,e^2=gi,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, 9758197);