#9266982

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	#51567	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3931 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)(5,6)(7,11)(10,15)(12,13)(14,16)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30);(2,6)(7,9)(8,13)(14,15)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32)(33,34,39,41)(35,42,36,37)(38,46,47,40)(43,45,44,48);(4,12)(8,13)(10,16)(14,15)(17,19,21,27)(18,22,24,25)(20,26,29,28)(23,30,32,31)(33,35,39,36)(34,37,41,42)(38,43,47,44)(40,45,46,48);(1,3,5,11)(2,7,6,9)(4,10,12,16)(8,14,13,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,36,39,35)(34,37,41,42)(38,44,47,43)(40,45,46,48);(1,4)(2,8)(3,10)(5,12)(6,13)(7,14)(9,15)(11,16)(17,20,21,29)(18,23,24,32)(19,26,27,28)(22,30,25,31)(33,37)(34,35)(36,41)(38,45)(39,42)(40,43)(44,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,11)(10,14)(12,13)(15,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,38)(34,40)(35,43)(36,44)(37,45)(39,47)(41,46)(42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,d^2=gi,e^2=h,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, 9174178);