#9245127

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

2^9|a^2=ghi,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, 9152323);