#478526

order	512
nilpotency class	3
isoclinism	isoclinic class of #478526
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#17152
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=def,g^b=gi,h^b=hi,d^c=de,g^c=gi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338618, #72338620, #72338629
maximal normal subgroup of	#72338618, #72338620, #72338629

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385722);