#231069

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄

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×#469	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

736 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 24 points:

(2,5)(6,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,23,21,19,20,24);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,15,12)(17,19,22,24,21,18,20,23);(1,3,4,7)(2,6,5,8)(9,12,16,10,13,14,11,15)(17,20,21,22)(18,23,19,24);(2,5)(6,8)(9,11,13,16)(10,14,15,12)

Polycyclic presentation:

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

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, 138265);