#8757280

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

3776 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:

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

Polycyclic presentation:

2^9|a^2=gi,c^2=hi,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664476);