#394680

order	512
nilpotency class	3
isoclinism	isoclinic class of #394680
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14718
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3021	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2130 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:

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

Polycyclic presentation:

2^9|c^2=e,e^2=i,f^2=i,h^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^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, 301876);