#9245126

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	#37471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18,21,23)(19,25,27,22)(20,24,29,31)(26,32,28,30)(33,34,38,41)(35,39,43,37)(36,44,46,40)(42,45,48,47);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,21,27)(18,22,23,25)(20,28,29,26)(24,32,31,30)(33,35,38,43)(34,37,41,39)(36,42,46,48)(40,45,44,47);(2,7)(3,10)(8,15)(11,12)(18,23)(20,29)(22,25)(26,28)(33,35)(34,39)(36,42)(37,41)(38,43)(40,47)(44,45)(46,48);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,18)(19,22)(20,24)(21,23)(25,27)(26,30)(28,32)(29,31)(33,36,38,46)(34,40,41,44)(35,42,43,48)(37,45,39,47);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,20,21,29)(18,24,23,31)(19,26,27,28)(22,30,25,32);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,29)(31,32)(33,37,38,39)(34,35,41,43)(36,45,46,47)(40,42,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,d^2=gh,e^2=h,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, 9152322);