#9720518

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	#45707	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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)(17,18,22,23)(19,25,30,24)(20,26,31,29)(21,28,32,27)(33,34,38,40)(35,36,42,44)(37,39,47,46)(41,43,45,48);(1,2)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(18,23)(20,31)(24,25)(27,28)(33,35)(34,36)(37,45)(38,42)(39,48)(40,44)(41,47)(43,46);(2,6)(7,9)(8,12)(14,15)(17,19,22,30)(18,24,23,25)(20,28,31,27)(21,29,32,26)(33,36,38,44)(34,35,40,42)(37,43,47,48)(39,41,46,45);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,19)(18,25)(20,28)(21,29)(22,30)(23,24)(26,32)(27,31)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,46)(47,48);(1,3,5,11)(2,7,6,9)(4,10,13,16)(8,14,12,15)(17,20)(18,26)(19,28)(21,24)(22,31)(23,29)(25,32)(27,30)(33,37)(34,39)(35,41)(36,43)(38,47)(40,46)(42,45)(44,48);(1,4)(2,8)(3,10)(5,13)(6,12)(7,14)(9,15)(11,16)(17,21,22,32)(18,27,23,28)(19,29,30,26)(20,25,31,24)(33,34)(35,36)(37,46)(38,40)(39,47)(41,48)(42,44)(43,45)

Polycyclic presentation:

2^9|a^2=h,c^2=gi,e^2=ghi,f^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, 9627714);