#9720535

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	#39728	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3970 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,24)(19,20,26,28)(21,23,31,30)(25,27,29,32)(36,44)(40,47)(43,45)(46,48);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,19)(18,20)(21,29)(22,26)(23,32)(24,28)(25,31)(27,30)(33,34)(35,42)(36,40)(37,39)(38,41)(43,48)(44,47)(45,46);(1,2,5,8)(3,6,11,9)(4,7,13,16)(10,14,12,15)(17,20,22,28)(18,19,24,26)(21,27,31,32)(23,25,30,29)(33,35,38,37)(34,39,41,42)(36,43,44,45)(40,46,47,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,30)(31,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,46)(41,47)(42,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,21)(18,23)(19,25)(20,27)(22,31)(24,30)(26,29)(28,32)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48);(1,3)(2,6)(4,12)(5,11)(7,15)(8,9)(10,13)(14,16)(17,18)(19,20)(21,30)(22,24)(23,31)(25,32)(26,28)(27,29)(33,37,38,35)(34,39,41,42)(36,45,44,43)(40,46,47,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=gi,e^2=g,f^2=ghi,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, 9627731);