#453859

order	512
nilpotency class	3
isoclinism	isoclinic class of #427666
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2527
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16160
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

570 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361055);