#8757328

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	#47575	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3901 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)(3,9)(4,7)(5,8)(6,12)(10,14)(11,15)(13,16)(17,18)(19,24)(20,22)(21,30)(23,25)(26,29)(27,32)(28,31)(33,34,39,36)(35,38,43,40)(37,41,45,44)(42,46,48,47);(1,3,7,5)(2,8,4,9)(6,13,14,11)(10,16,12,15)(17,19)(18,24)(20,29)(21,27)(22,26)(23,28)(25,31)(30,32)(37,45)(41,44)(42,48)(46,47);(2,4)(8,9)(10,12)(15,16)(18,25)(19,28)(20,27)(26,30)(33,35,39,43)(34,40,36,38)(37,42,45,48)(41,47,44,46);(1,4,7,2)(3,9,5,8)(6,12,14,10)(11,16,13,15)(17,20,23,27)(18,22,25,32)(19,29,28,21)(24,26,31,30)(33,36,39,34)(35,38,43,40)(37,44,45,41)(42,46,48,47);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,21,23,29)(18,26,25,30)(19,27,28,20)(22,31,32,24)(33,37)(34,41)(35,42)(36,44)(38,46)(39,45)(40,47)(43,48);(1,6,7,14)(2,10,4,12)(3,11,5,13)(8,15,9,16)(17,22)(18,27)(19,30)(20,25)(21,31)(23,32)(24,29)(26,28)(33,38)(34,35)(36,43)(37,46)(39,40)(41,42)(44,48)(45,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=h,e^2=gi,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, 8664524);