#231475

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2366
prime radical	C₂×#493	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

732 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=f,d^2=eh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,f^a=fh,c^b=ci,d^b=de,e^b=eh,g^b=gh

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, 138671);