#350949

order	512
nilpotency class	3
isoclinism	isoclinic class of #347668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

614 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)(4,5)(7,8)(10,13)(11,14)(15,16)(17,18,22,24)(19,25,26,32)(20,21,28,29)(23,30,31,27);(2,5)(6,8)(17,19,21,27,22,26,29,30)(18,23,20,25,24,31,28,32);(1,3,4,7)(2,6,5,8)(9,10,12,13)(11,14,15,16)(17,20,21,24,22,28,29,18)(19,25,27,31,26,32,30,23);(9,11,12,15)(10,14,13,16)(19,26)(23,31)(25,32)(27,30)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=ei,d^2=i,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fh,g^a=gi,d^b=dh,g^d=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 258145);