#9850634

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

4108 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gi,e^2=gi,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, 9757830);