#435321

order	512
nilpotency class	3
isoclinism	isoclinic class of #420065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2686
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15524
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

615 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)(3,5)(4,10,21,28)(6,13)(7,14)(8,12)(9,16)(11,23)(15,30,26,19)(17,22)(18,32,31,25)(20,29,24,27);(1,3,8,16,7,11,6,17)(2,9,12,23,14,22,13,5)(4,18,24,26,21,31,20,15)(10,19,27,25,28,30,29,32);(1,4,8,24,7,21,6,20)(2,10,12,27,14,28,13,29)(3,15,16,18,11,26,17,31)(5,19,9,25,23,30,22,32);(1,5,6,22,7,23,8,9)(2,11,12,17,14,3,13,16)(4,19,20,32,21,30,24,25)(10,26,27,31,28,15,29,18)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=ef,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,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, 342517);