#8455688

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	#54866	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4015 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|b^2=ghi,c^2=ghi,d^2=gh,e^2=hi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8362884);