#8127607

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3964 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=i,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8034803);