#8728915

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	#53390	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4120 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(18,22)(19,25)(24,27)(26,28)(33,34,38,41)(35,36,43,39)(37,40,47,48)(42,44,45,46);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(18,22)(23,30)(24,27)(31,32)(33,35,38,43)(34,39,41,36)(37,42,47,45)(40,46,48,44);(3,10)(6,7)(11,16)(14,15)(17,18,21,22)(19,23,25,30)(20,24,29,27)(26,31,28,32)(33,35,38,43)(34,39,41,36)(37,42,47,45)(40,46,48,44);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35)(34,36)(37,45)(38,43)(39,41)(40,46)(42,47)(44,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,19)(18,23)(20,28)(21,25)(22,30)(24,32)(26,29)(27,31)(33,36)(34,35)(37,46)(38,39)(40,45)(41,43)(42,48)(44,47);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,20,21,29)(18,24,22,27)(19,26,25,28)(23,31,30,32)(33,37)(34,40)(35,42)(36,44)(38,47)(39,46)(41,48)(43,45)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8636111);