#9266997

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	#41317	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3853 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(33,34,38,39)(35,40,42,37)(36,44,46,41)(43,45,48,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,18,23,25)(19,26,20,21)(22,30,31,24)(27,29,28,32)(34,39)(35,42)(41,44)(43,48);(2,3)(7,9)(8,10)(14,15)(17,19,23,20)(18,21,25,26)(22,27,31,28)(24,29,30,32)(33,35)(34,40)(36,43)(37,39)(38,42)(41,47)(44,45)(46,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,19)(18,21,25,26)(22,28,31,27)(24,29,30,32)(33,36,38,46)(34,41,39,44)(35,43,42,48)(37,45,40,47);(1,4)(2,7)(3,9)(5,11)(6,12)(8,14)(10,15)(13,16)(17,21)(18,19)(20,25)(22,29)(23,26)(24,27)(28,30)(31,32)(33,37,38,40)(34,35,39,42)(36,45,46,47)(41,43,44,48);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22)(18,24)(19,27)(20,28)(21,29)(23,31)(25,30)(26,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gi,d^2=i,e^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, 9174193);