#6431726

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431723
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

7054 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)(7,17)(8,13)(9,14)(10,24)(11,16)(12,19)(15,25)(18,31)(20,28)(21,27)(22,29)(23,35)(26,32)(30,34)(33,36);(5,7,11,8)(6,12,16,27)(9,18,22,20)(10,21,25,19)(13,24,17,15)(14,26,29,36)(23,33,34,32)(28,35,31,30);(2,4)(5,8,11,7)(6,13,16,17)(9,20,22,18)(10,19,25,21)(12,15,27,24)(14,28,29,31)(23,32,34,33)(26,30,36,35);(5,9)(6,14)(7,18)(8,20)(10,23)(11,22)(12,26)(13,28)(15,30)(16,29)(17,31)(19,32)(21,33)(24,35)(25,34)(27,36);(9,22)(14,29)(18,20)(23,34)(26,36)(28,31)(30,35)(32,33)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306978, #72307024, #72307032
maximal normal subgroup of	#72306978, #72307024, #72307032

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6338922);