#8461524

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	#46042	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|a^2=gh,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, 8368720);