#8461582

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

4313 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=gh,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, 8368778);