#7654539

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4531 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^2=gh,f^2=h,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, 7561735);