#8464286

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	#35044	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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)(3,12,14,9)(4,10,5,15)(6,16,13,11)(17,18,22,24)(19,25,27,21)(20,23,29,30)(26,31,32,28)(33,34)(35,39)(36,46)(37,42)(38,43)(40,44)(41,45)(47,48);(1,3)(2,9)(4,6)(5,13)(7,12)(8,14)(10,11)(15,16)(17,19,22,27)(18,21,24,25)(20,26,29,32)(23,28,30,31)(33,35,40,38)(34,39,44,43)(36,47,45,37)(41,42,46,48);(1,4,8,5)(2,10,7,15)(3,6,14,13)(9,11,12,16)(17,18)(19,21)(20,23)(22,24)(25,27)(26,28)(29,30)(31,32)(33,36)(34,41)(35,37)(38,47)(39,48)(40,45)(42,43)(44,46);(1,5,8,4)(2,10,7,15)(3,13,14,6)(9,11,12,16)(17,18)(19,25)(20,23)(21,27)(22,24)(26,31)(28,32)(29,30)(33,37,40,47)(34,42,44,48)(35,45,38,36)(39,41,43,46);(1,6,8,13)(2,11,7,16)(3,4,14,5)(9,10,12,15)(17,20,22,29)(18,23,24,30)(19,26,27,32)(21,28,25,31)(33,38,40,35)(34,43,44,39)(36,47,45,37)(41,42,46,48);(1,7,8,2)(3,12,14,9)(4,15,5,10)(6,16,13,11)(17,21,22,25)(18,19,24,27)(20,28,29,31)(23,26,30,32)(33,39,40,43)(34,35,44,38)(36,48,45,42)(37,46,47,41)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=gh,e^2=gi,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, 8371482);