#353357

order	512
nilpotency class	3
isoclinism	isoclinic class of #3068
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4098 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238745, #72238748, #72238755, …
maximal normal subgroup of	#72238745, #72238748, #72238755, …
maximal quotient of	#72239083

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260553);