#7646754

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	#2656	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4284 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)(11,16)(13,21)(19,24)(25,26)(27,29)(28,36)(30,33)(31,35)(32,38)(34,39)(37,40);(1,2)(3,6)(4,5)(7,8)(9,10,14,15)(11,12,16,20)(13,17,21,22)(18,19,23,24)(25,27,30,35)(26,29,33,31)(28,34,38,40)(32,37,36,39);(1,3,4,7)(2,6,5,8)(9,10,14,15)(11,12,16,20)(13,17,21,22)(18,19,23,24)(25,27)(26,31)(28,34)(29,33)(30,35)(32,39)(36,37)(38,40);(9,11)(10,12)(13,18)(14,16)(15,20)(17,19)(21,23)(22,24)(25,28,30,38)(26,32,33,36)(27,34,35,40)(29,37,31,39);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,29,30,31)(26,27,33,35)(28,37,38,39)(32,34,36,40);(9,13)(10,17)(11,18)(12,19)(14,21)(15,22)(16,23)(20,24)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7553950);