#8757409

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	#56279	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3903 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)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,34,39,37)(35,40,45,46)(36,42,38,41)(43,48,44,47);(2,6)(4,12)(7,9)(11,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(33,35)(34,40)(36,43)(37,46)(38,44)(39,45)(41,47)(42,48);(3,10)(7,9)(11,16)(14,15)(18,22)(23,25)(24,28)(30,31)(33,36,39,38)(34,41,37,42)(35,43,45,44)(40,47,46,48);(1,3,5,10)(2,7,6,9)(4,11,12,16)(8,14,13,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,37,39,34)(35,40,45,46)(36,42,38,41)(43,47,44,48);(1,4,5,12)(2,8,6,13)(3,11,10,16)(7,14,9,15)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)(33,38,39,36)(34,42,37,41)(35,44,45,43)(40,48,46,47)

Polycyclic presentation:

2^9|a^2=ghi,c^2=hi,d^2=g,e^2=hi,f^2=gh,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, 8664605);