#8127403

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4034 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)(3,8)(4,6)(5,12)(7,10)(9,13)(11,14)(15,16)(17,18,21,24)(19,22,26,31)(20,27,29,23)(25,32,28,30)(37,45)(39,46)(42,48)(44,47);(3,10)(7,8)(11,16)(14,15)(17,18)(19,22)(20,23)(21,24)(25,30)(26,31)(27,29)(28,32)(33,34,38,40)(35,41,43,36)(37,39,45,46)(42,47,48,44);(1,3,6,10)(2,7,4,8)(5,11,13,16)(9,14,12,15)(17,19,21,26)(18,22,24,31)(20,28,29,25)(23,32,27,30)(33,35,38,43)(34,36,40,41)(37,42,45,48)(39,44,46,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,14)(13,16)(17,20,21,29)(18,23,24,27)(19,25,26,28)(22,30,31,32)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,4,6,2)(3,7,10,8)(5,12,13,9)(11,14,16,15)(17,19)(18,22)(20,25)(21,26)(23,30)(24,31)(27,32)(28,29)(33,36)(34,35)(37,44)(38,41)(39,42)(40,43)(45,47)(46,48);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,18)(19,22)(20,23)(21,24)(25,30)(26,31)(27,29)(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=hi,c^2=gi,d^2=g,e^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, 8034599);