#7654492

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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4426 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)(3,6,4,7)(10,11)(15,17)(16,18)(23,24)(25,26)(27,32)(28,33)(29,34)(30,31)(35,36)(37,39)(38,40);(1,2)(3,6)(4,7)(5,8)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24)(26,31)(27,34)(33,39)(35,40);(1,3,5,4)(2,6,8,7)(9,10,14,11)(12,15,20,17)(13,16,22,18)(19,23,21,24)(25,27,30,34)(26,32,31,29)(28,35,37,40)(33,36,39,38);(9,11,14,10)(12,17,20,15)(13,18,22,16)(19,24,21,23);(9,12,14,20)(10,15,11,17)(13,21,22,19)(16,24,18,23)(25,28)(26,33)(27,35)(29,38)(30,37)(31,39)(32,36)(34,40);(1,4,5,3)(2,7,8,6)(9,13,14,22)(10,16,11,18)(12,19,20,21)(15,23,17,24)(25,29,30,32)(26,34,31,27)(28,36,37,38)(33,35,39,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=g,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7561688);