#357633

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240013, #72240100
maximal normal subgroup of	#72240013, #72240100
maximal quotient of	#72235046, #72240021

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264829);