#475426

order	512
nilpotency class	3
isoclinism	isoclinic class of #460462
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2499
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17048
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

590 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:

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

Polycyclic presentation:

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

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, 382622);