#9711964

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	#55514	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=hi,d^2=g,e^2=gh,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,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, 9619160);