#7656560

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4397 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,c^b=ci,f^e=fhi

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, 7563756);