#231129

order	512
nilpotency class	3
isoclinism	isoclinic class of #23848
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

646 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)(11,18)(12,15)(13,17)(14,20)(16,19)(21,22)(23,24)(25,26,31,35,30,34,29,33)(27,36,37,40,38,32,28,39);(1,2)(3,6)(4,5)(7,8)(25,27,31,37,30,38,29,28)(26,32,35,39,34,36,33,40);(1,3,4,7)(2,6,5,8)(9,11,19,24,13,21,12,20)(10,14,15,22,17,23,16,18)(25,28,29,38,30,37,31,27)(26,32,35,39,34,36,33,40);(2,5)(6,8)(9,12,13,19)(10,15,17,16)(11,20,21,24)(14,22,23,18)(25,29,30,31)(26,33,34,35)(27,37,38,28)(32,39,36,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=efh,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,d^b=de,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 138325);