#409721

order	512
nilpotency class	3
isoclinism	isoclinic class of #402003
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2494
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15051
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

563 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)(3,12)(4,16,18,8)(5,9)(6,10)(7,14)(11,20)(13,26,17,21)(15,23)(19,25,27,24)(22,29)(28,32,30,31);(1,3,5,15,6,14,20,29)(2,7,11,23,10,12,9,22)(4,17,19,30,18,13,27,28)(8,21,25,32,16,26,24,31);(1,4)(2,8)(3,13,14,17)(5,19)(6,18)(7,21,12,26)(9,24)(10,16)(11,25)(15,28,29,30)(20,27)(22,31,23,32);(2,9,10,11)(7,22,12,23)(8,24,16,25)(21,31,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316917);