#6431928

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431785
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#577	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4574 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339124);