#7630655

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×#585	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

22314 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537851);