#9267022

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	#57153	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3852 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)(3,5)(4,9)(6,7)(8,11)(10,13)(12,14)(15,16)(17,18,22,23)(19,25,27,20)(21,29,31,24)(26,28,30,32);(2,7)(5,8)(9,14)(13,15)(17,19,22,27)(18,20,23,25)(21,30,31,26)(24,32,29,28)(33,34,39,41)(35,42,36,37)(38,46,47,40)(43,45,44,48);(4,12)(9,14)(10,16)(13,15)(18,23)(20,25)(24,29)(28,32)(33,35,39,36)(34,37,41,42)(38,43,47,44)(40,45,46,48);(1,3)(2,8)(4,10)(5,7)(6,11)(9,15)(12,16)(13,14)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)(29,31)(30,32)(33,36,39,35)(34,37,41,42)(38,44,47,43)(40,45,46,48);(1,4,6,12)(2,9,7,14)(3,10,11,16)(5,13,8,15)(17,20)(18,19)(21,28)(22,25)(23,27)(24,26)(29,30)(31,32)(33,37)(34,35)(36,41)(38,45)(39,42)(40,43)(44,46)(47,48);(1,5,6,8)(2,3,7,11)(4,13,12,15)(9,10,14,16)(17,21,22,31)(18,24,23,29)(19,26,27,30)(20,28,25,32)(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=i,e^2=gh,f^2=g,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, 9174218);