#9266992

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	#42760	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3865 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(33,34,38,39)(35,41,44,40)(36,45,47,37)(42,43,46,48);(1,2,5,8)(3,6,11,9)(4,7,12,14)(10,13,16,15)(17,18,23,25)(19,26,20,21)(22,30,31,24)(27,29,28,32)(34,39)(37,45)(40,41)(43,48);(4,12)(7,14)(10,16)(13,15)(17,19,23,20)(18,21,25,26)(22,27,31,28)(24,29,30,32)(33,35)(34,40)(36,46)(37,48)(38,44)(39,41)(42,47)(43,45);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(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)(39,45)(40,43)(41,48)(44,46);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,21)(18,19)(20,25)(22,29)(23,26)(24,27)(28,30)(31,32)(33,37)(34,36)(35,43)(38,45)(39,47)(40,42)(41,46)(44,48);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,14)(15,16)(17,22)(18,24)(19,27)(20,28)(21,29)(23,31)(25,30)(26,32)(33,34)(35,40)(36,37)(38,39)(41,44)(42,43)(45,47)(46,48)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,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, 9174188);