#9719174

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3869 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=g,d^2=g,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, 9626370);