#357826

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

2946 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)(9,10,15,18)(11,19,21,24)(12,20,22,16)(13,17,14,23);(9,11,14,16,15,21,13,20)(10,12,17,19,18,22,23,24);(2,4)(9,12)(10,16)(11,17)(13,19)(14,24)(15,22)(18,20)(21,23);(9,13,15,14)(10,17,18,23)(11,20,21,16)(12,19,22,24)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239938
maximal normal subgroup of	#72239938
maximal quotient of	#72239944

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265022);