#7662427

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	#37280	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4303 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=h,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,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, 7569623);