#123546

order	512
nilpotency class	3
isoclinism	isoclinic class of #8325
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂ ▷ C₁

2646 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,b^a=bd,c^a=ce,d^a=di,c^b=cf,g^b=gi,e^c=ei,g^c=gi

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, 30742);