#9240942

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	#51922	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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)(3,9)(4,8)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18)(19,22)(20,23)(21,26)(24,29)(25,28)(27,31)(30,32)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(2,6)(7,9)(8,13)(14,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(35,43)(37,47)(39,48)(41,44);(4,12)(8,13)(10,16)(14,15)(18,19)(23,25)(24,26)(30,31)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,4,5,12)(2,8,6,13)(3,10,11,16)(7,14,9,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,d^2=h,e^2=gh,f^2=h,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, 9148138);