#9478358

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	#47026	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9385554);