#357718

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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239937, #72240100
maximal normal subgroup of	#72239937, #72240100
maximal quotient of	#72235046, #72239944

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264914);