#8143156

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	#53527	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3905 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=ghi,d^2=gi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8050352);