#373629

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

781 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=efh,c^2=ef,d^2=efg,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi,e^d=ei,f^d=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 280825);