#9717596

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	#49921	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4080 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)(3,5)(4,8)(6,7)(9,13)(10,12)(11,14)(15,16)(17,18)(19,26)(20,24)(21,29)(22,23)(25,31)(27,28)(30,32)(33,34)(35,43)(36,37)(38,41)(39,42)(40,46)(44,48)(45,47);(4,11)(8,14)(10,15)(12,16)(17,18)(19,20)(21,25)(22,23)(24,26)(27,28)(29,31)(30,32)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,48)(42,43);(2,7)(3,9)(8,14)(10,15)(18,23)(19,24)(25,29)(27,32)(33,36,39,38)(34,41,42,37)(35,44,46,47)(40,48,43,45);(1,3,6,9)(2,5,7,13)(4,10,11,15)(8,12,14,16)(17,19,22,24)(18,20,23,26)(21,27,31,32)(25,28,29,30)(33,37,39,41)(34,36,42,38)(35,45,46,48)(40,44,43,47);(1,4,6,11)(2,8,7,14)(3,10,9,15)(5,12,13,16)(17,20)(18,24)(19,23)(21,30)(22,26)(25,27)(28,31)(29,32)(33,38,39,36)(34,41,42,37)(35,44,46,47)(40,45,43,48);(1,5)(2,9)(3,7)(4,12)(6,13)(8,15)(10,14)(11,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,29)(24,32)(26,30)(33,35)(34,40)(36,44)(37,45)(38,47)(39,46)(41,48)(42,43)

Polycyclic presentation:

2^9|c^2=g,d^2=h,e^2=hi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9624792);