#357776

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2902 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)(3,4)(5,6,7,8)(9,10,15,19)(11,17,12,23)(13,18,14,20)(16,21,24,22);(9,11,14,22,15,12,13,21)(10,16,18,17,19,24,20,23);(2,4)(9,12)(10,17)(11,15)(13,22)(14,21)(16,20)(18,24)(19,23);(9,13,15,14)(10,18,19,20)(11,21,12,22)(16,17,24,23)

Polycyclic presentation:

2^9|a^2=h,b^2=e,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=ei,c^b=cg,g^b=gi,g^c=gi

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, 264972);