#228376

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³

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

603 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=dehi,d^c=dh,f^c=fh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 135572);