#9481447

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	#37228	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3773 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=hi,e^2=g,f^2=h,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, 9388643);