#555898

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂²×#525	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5710 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=i,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327959, #72327961, #72327972
maximal normal subgroup of	#72327959, #72327961, #72327972
maximal quotient of	#72335941

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 463094);