#8459128

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	#2537	C₂²
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) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=gh,e^2=ghi,f^2=gi,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, 8366324);