#8757300

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	#56916	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3837 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=h,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, 8664496);