#228327

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	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

(2,5)(6,8)(9,10)(11,15)(12,17)(13,16)(14,18)(19,22)(20,24)(21,23)(25,26,30,34,31,28,29,33)(27,32,36,39,37,35,38,40);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,22)(17,23)(18,24)(25,27,30,36,31,37,29,38)(26,32,34,39,28,35,33,40);(1,3,4,7)(2,6,5,8)(10,16)(12,13)(15,22)(17,18)(19,21)(23,24)(25,28)(26,31)(27,35)(29,34)(30,33)(32,37)(36,40)(38,39);(2,5)(6,8)(9,12,14,13)(10,17,18,16)(11,19,20,21)(15,22,24,23)(25,29,31,30)(26,33,28,34)(27,36,37,38)(32,39,35,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=deh,d^c=dhi,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, 135523);