#8755407

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	#50460	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3835 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=ghi,d^2=ghi,e^2=h,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, 8662603);