#9266991

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	#56422	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3851 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=i,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, 9174187);