#8455687

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	#55915	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 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)(3,11)(4,9)(5,6)(7,10)(8,15)(12,14)(13,16)(18,23)(19,28)(26,29)(27,32)(34,38)(39,45)(40,46)(47,48);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,18,22,23)(19,24,28,25)(20,26,30,29)(21,27,31,32)(34,38)(35,41)(40,46)(42,43);(1,4,7,5)(2,9,10,6)(3,12,15,13)(8,16,11,14)(17,19)(18,24)(20,27)(21,29)(22,28)(23,25)(26,31)(30,32)(33,34)(35,39)(36,40)(37,38)(41,45)(42,47)(43,48)(44,46);(1,5,7,4)(2,9,10,6)(3,13,15,12)(8,16,11,14)(17,19)(18,25)(20,27)(21,26)(22,28)(23,24)(29,31)(30,32)(33,35,37,41)(34,39,38,45)(36,43,44,42)(40,48,46,47);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,20,22,30)(18,26,23,29)(19,27,28,32)(21,24,31,25);(1,6,7,9)(2,5,10,4)(3,14,15,16)(8,13,11,12)(17,21)(18,27)(19,29)(20,24)(22,31)(23,32)(25,30)(26,28)(33,36,37,44)(34,40,38,46)(35,42,41,43)(39,47,45,48)

Polycyclic presentation:

2^9|b^2=hi,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, 8362883);