#400224

order	512
nilpotency class	3
isoclinism	isoclinic class of #394483
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2472
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14797
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

884 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)(4,16,19,28)(5,22)(6,11)(7,12)(9,20,27,21)(10,23)(13,25)(14,18,17,31)(15,24)(26,29,32,30);(1,3)(2,8)(4,17,19,14)(5,13)(6,15)(7,10)(9,26,27,32)(11,24)(12,23)(16,18,28,31)(20,30,21,29)(22,25);(1,4)(2,9)(3,14)(5,18)(6,21)(7,19)(8,26)(10,17)(11,28)(12,27)(13,16)(15,30)(20,25)(22,29)(23,32)(24,31);(1,5,6,8,7,24,25,23)(2,10,11,22,12,3,13,15)(4,18,21,26,19,31,20,32)(9,17,28,29,27,14,16,30)

Polycyclic presentation:

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

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, 307420);