#8461523

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4138 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=h,e^2=ghi,f^2=i,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, 8368719);