#8430757

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	#38564	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4567 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,d^2=g,e^2=gh,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, 8337953);