#6443130

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3500	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5022 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=f,c^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6350326);