#8094469

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	#45701	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4459 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8001665);