#313711

order	512
nilpotency class	3
isoclinism	isoclinic class of #264123
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

721 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)(3,6)(4,5)(7,8)(10,11)(12,16)(13,14)(17,18)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31);(2,5)(6,8)(9,10,14,16,15,12,13,11)(17,19,20,26)(18,22,23,30)(21,25,28,27)(24,32,31,29);(1,3,4,7)(2,6,5,8)(9,11,14,10,15,16,13,12)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(2,5)(6,8)(9,12,14,11,15,10,13,16)

Polycyclic presentation:

2^9|b^2=f,c^2=eh,d^2=fi,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,g^b=gi,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, 220907);