#7607476

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518904
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1766 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fgi,c^2=ghi,d^2=i,e^2=fgi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282893, #72283173, #72284559
maximal normal subgroup of	#72282893, #72283173, #72284559

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7514672);