#508541

order	512
nilpotency class	4
isoclinism	isoclinic class of #508540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#29925
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

477 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415737);