#455255

order	512
nilpotency class	3
isoclinism	isoclinic class of #455254
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16261
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1267 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2,8,12,7,13,6,22)(3,14,23,26,18,29,17,9)(4,10,21,27,20,28,5,11)(15,25,16,24,19,31,30,32);(1,3,6,17,7,18,8,23)(2,9,12,14,13,26,22,29)(4,19,5,16,20,15,21,30)(10,24,27,31,28,32,11,25);(1,4)(2,10)(3,15)(5,8)(6,21)(7,20)(9,24)(11,12)(13,28)(14,25)(16,17)(18,19)(22,27)(23,30)(26,32)(29,31);(1,5,8,4,7,21,6,20)(2,11,22,28,13,27,12,10)(3,16,23,19,18,30,17,15)(9,25,29,32,26,31,14,24)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362451);