#6442586

order	512
nilpotency class	3
isoclinism	isoclinic class of #6442584
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4542 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286950, #72286962, #72287089
maximal normal subgroup of	#72286950, #72286962, #72287089

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349782);