#9839328

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	#41962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4153 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)(3,9,11,4)(5,8,14,15)(10,13,16,12)(17,18)(19,21)(20,24)(22,23)(25,29)(26,28)(27,30)(31,32)(33,34,39,36)(35,37,40,42)(38,44,47,41)(43,46,48,45);(2,7)(4,9)(8,15)(12,13)(20,27)(24,30)(26,31)(28,32)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(18,23)(19,25)(24,30)(26,31)(34,36)(35,40)(41,44)(43,48);(1,3)(2,4)(5,10)(6,11)(7,9)(8,12)(13,15)(14,16)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,36,39,34)(35,42,40,37)(38,44,47,41)(43,46,48,45);(1,4)(2,3)(5,13)(6,9)(7,11)(8,16)(10,15)(12,14)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37,39,42)(34,40,36,35)(38,46,47,45)(41,43,44,48);(1,5,6,14)(2,8,7,15)(3,10,11,16)(4,12,9,13)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

2^9|a^2=gi,d^2=h,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, 9746524);