#501334

order	512
nilpotency class	4
isoclinism	isoclinic class of #501334
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24506
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

669 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408530);