#8438905

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	#37748	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4599 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:

(1,2,6,5)(3,4,7,8)(10,14)(11,17)(16,19)(18,23)(28,36)(32,39)(34,40)(37,38);(1,3,6,7)(2,4,5,8)(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24)(25,26)(27,31)(28,32)(29,33)(30,35)(34,38)(36,39)(37,40);(1,4)(2,3)(5,7)(6,8)(25,27)(26,30)(28,34)(29,35)(31,33)(32,37)(36,40)(38,39);(1,5,6,2)(3,4,7,8)(9,10)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23)(25,27)(26,31)(28,34)(29,35)(30,33)(32,38)(36,40)(37,39);(9,11,13,17)(10,15,14,21)(12,18,20,23)(16,22,19,24)(25,28,29,36)(26,32,33,39)(27,34,35,40)(30,37,31,38);(1,4)(2,7)(3,5)(6,8)(9,12,13,20)(10,16,14,19)(11,18,17,23)(15,22,21,24)(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,d^2=h,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, 8346101);