#326024

order	512
nilpotency class	3
isoclinism	isoclinic class of #288899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

592 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:

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

Polycyclic presentation:

2^9|b^2=eg,c^2=ei,d^2=fhi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,f^a=fh,c^b=cg,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233220);