#359657

order	512
nilpotency class	3
isoclinism	isoclinic class of #24464
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3489	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

4036 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)(9,10,14,11,15,17,13,19)(12,20,21,24,22,18,23,16);(5,6)(7,8)(9,11)(10,13)(12,18)(14,17)(15,19)(16,21)(20,22)(23,24);(2,4)(6,8)(9,12)(10,16)(11,18)(13,21)(14,23)(15,22)(17,24)(19,20);(9,10,15,17)(11,13,19,14)(12,16,22,24)(18,21,20,23)

Polycyclic presentation:

2^9|a^2=h,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^c=fi,h^c=hi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316872
maximal normal subgroup of	#72316872
maximal quotient of	#72317090

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266853);