#99528

order	512
nilpotency class	3
isoclinism	isoclinic class of #10046
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	#386
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

470 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=g,d^2=g,e^2=h,f^2=i,b^a=bd,c^a=ce,c^b=ci,d^b=dg,e^b=eh,f^b=fg,d^c=dh,e^c=eh,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, 6724);