#382902

order	512
nilpotency class	3
isoclinism	isoclinic class of #382477
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2310
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14346
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

839 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)(3,12,15,22)(4,9)(5,10)(6,20)(7,11)(8,13,24,21)(14,28,16,30)(17,26)(18,32)(19,27)(23,29,25,31);(3,13,15,21)(8,22,24,12)(14,29,16,31)(23,30,25,28);(1,3)(2,8,11,24)(4,16,19,14)(5,13)(6,21)(7,15)(9,25)(10,12,20,22)(17,31,18,29)(23,27)(26,30)(28,32);(1,4,5,17,7,19,6,18)(2,9,10,26,11,27,20,32)(3,14,13,29,15,16,21,31)(8,23,12,28,24,25,22,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 290098);