#342669

order	512
nilpotency class	3
isoclinism	isoclinic class of #334823
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

739 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)(3,7)(4,6)(5,8)(9,10,13,14,12,16,11,15)(19,26)(23,31)(27,32)(28,30);(1,3)(2,5)(4,8)(6,7)(9,11,12,13)(10,14,16,15)(17,18,21,25)(19,27,26,32)(20,24,22,29)(23,30,31,28);(10,15)(11,13)(14,16)(17,19)(18,23)(20,28)(21,26)(22,30)(24,32)(25,31)(27,29);(2,6)(3,8)(10,14)(11,13)(15,16)(17,20,21,22)(18,24,25,29)(19,28,26,30)(23,32,31,27)

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 249865);