#8438533

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	#42480	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4699 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,d^2=g,e^2=ghi,f^2=g,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, 8345729);