#9267026

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3704 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dh,e^c=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9174222);