#7648452

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	#37554	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4464 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)(9,10,14,17)(11,12,19,15)(13,21,22,16)(18,23,24,20)(26,32)(30,34)(33,35)(37,40);(1,2,4,5)(3,6,7,8)(25,26,31,32)(27,35,29,33)(28,34,38,30)(36,37,39,40);(9,11)(10,15)(12,17)(13,18)(14,19)(16,23)(20,21)(22,24)(25,27,31,29)(26,33,32,35)(28,36,38,39)(30,37,34,40);(9,12,14,15)(10,11,17,19)(13,20,22,23)(16,18,21,24)(25,28)(26,34)(27,36)(29,39)(30,32)(31,38)(33,40)(35,37);(1,3,4,7)(2,6,5,8)(9,11)(10,12)(13,18)(14,19)(15,17)(16,20)(21,23)(22,24)(25,29,31,27)(26,33,32,35)(28,36,38,39)(30,40,34,37);(9,13,14,22)(10,16,17,21)(11,18,19,24)(12,20,15,23)(25,30,31,34)(26,28,32,38)(27,37,29,40)(33,36,35,39)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=h,e^2=g,f^2=hi,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, 7555648);