#362595

order	512
nilpotency class	3
isoclinism	isoclinic class of #362481
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2293
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#13919
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

839 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 269791);