#8806869

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	#53189	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4058 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=ghi,d^2=hi,e^2=hi,f^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=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8714065);