#459762

order	512
nilpotency class	3
isoclinism	isoclinic class of #455576
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2730
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16348
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

900 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366958);