#7674640

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

10702 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=ei

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, 7581836);