#7655259

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4450 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,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, 7562455);