#9245139

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

3721 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)(4,6)(5,12)(7,11)(9,13)(10,14)(15,16)(17,18,22,25)(19,20,23,28)(21,24,30,32)(26,27,29,31)(33,34)(35,42)(36,37)(38,40)(39,43)(41,48)(44,45)(46,47);(1,2,6,4)(3,7,11,8)(5,12,13,9)(10,15,16,14)(17,19,22,23)(18,20,25,28)(21,29,30,26)(24,31,32,27)(33,35,39,38)(34,40,43,42)(36,46,41,44)(37,45,48,47);(1,3,6,11)(2,7,4,8)(5,10,13,16)(9,14,12,15)(17,19)(18,20)(21,26)(22,23)(24,27)(25,28)(29,30)(31,32)(33,36,39,41)(34,37,43,48)(35,44,38,46)(40,47,42,45);(1,4,6,2)(3,7,11,8)(5,12,13,9)(10,14,16,15)(33,37)(34,41)(35,45)(36,43)(38,47)(39,48)(40,44)(42,46);(1,3)(2,8)(4,7)(5,10)(6,11)(9,15)(12,14)(13,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,29)(26,32)(30,31)(33,35,39,38)(34,42,43,40)(36,44,41,46)(37,45,48,47);(1,5)(2,9)(3,10)(4,12)(6,13)(7,14)(8,15)(11,16)(17,21)(18,24)(19,26)(20,27)(22,30)(23,29)(25,32)(28,31)(33,38,39,35)(34,40,43,42)(36,46,41,44)(37,47,48,45)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=g,d^2=h,e^2=gh,f^2=gh,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, 9152335);