#226521

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	C₂×C₄²	C₂⁴
socle, prime radical	C₂³	C₂×C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₄⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

618 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fh,d^2=h,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,d^b=de,e^b=eh,e^d=eh

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, 133717);