#8127466

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

3844 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=hi,d^2=g,e^2=h,f^2=g,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, 8034662);