#394822

order	512
nilpotency class	3
isoclinism	isoclinic class of #394821
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14719
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3068	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1560 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)(3,8)(4,9)(5,14)(6,12)(7,11)(10,20)(13,18)(15,19)(16,27)(17,24)(21,26)(22,25)(23,29)(28,30)(31,32);(1,3,8,2,7,12,6,11)(4,15,18,13,19,9,17,24)(5,14,22,10,21,26,20,25)(16,28,30,27,31,23,29,32);(1,4,6,17,7,19,8,18)(2,9,3,13,11,15,12,24)(5,16,20,29,21,31,22,30)(10,23,14,27,25,28,26,32);(1,5)(2,10)(3,14)(4,16)(6,20)(7,21)(8,22)(9,23)(11,25)(12,26)(13,27)(15,28)(17,29)(18,30)(19,31)(24,32)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 302018);