#7625722

order	512
nilpotency class	2
isoclinism	isoclinic class of #62842
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

10626 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)(9,10,16,19)(11,13,12,22)(14,17,28,33)(15,29,30,18)(20,25,23,31)(21,35,24,26)(27,38,39,32)(34,40,36,37);(5,6)(7,8)(9,11,16,12)(10,13,19,22)(14,23,28,20)(15,21,30,24)(17,31,33,25)(18,26,29,35)(27,36,39,34)(32,40,38,37);(1,3,4,2)(9,12,16,11)(10,13,19,22)(14,23,28,20)(15,24,30,21)(17,25,33,31)(18,26,29,35)(27,36,39,34)(32,37,38,40);(9,13)(10,12)(11,19)(14,25)(15,26)(16,22)(17,23)(18,24)(20,33)(21,29)(27,37)(28,31)(30,35)(32,36)(34,38)(39,40);(9,14,16,28)(10,17,19,33)(11,20,12,23)(13,25,22,31)(15,27,30,39)(18,32,29,38)(21,34,24,36)(26,37,35,40);(9,15)(10,18)(11,21)(12,24)(13,26)(14,27)(16,30)(17,32)(19,29)(20,34)(22,35)(23,36)(25,37)(28,39)(31,40)(33,38)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=i,e^2=h,b^a=bg,c^a=ch,f^a=fh,e^b=eh,d^c=dh

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532918);