#473198

order	512
nilpotency class	3
isoclinism	isoclinic class of #456381
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2712
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16853
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

561 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)(3,13,16,5)(4,9)(6,10)(7,12)(8,15,23,17)(11,24)(14,22,31,26)(18,19,30,25)(20,27)(21,29)(28,32);(1,3)(2,8)(4,18,21,30)(5,10)(6,17)(7,16)(9,26,29,22)(11,13)(12,23)(14,20,31,32)(15,24)(19,27,25,28);(1,4,6,20,7,21,24,32)(2,9,10,27,12,29,11,28)(3,14,15,30,16,31,17,18)(5,22,8,25,13,26,23,19);(1,5,6,23,7,13,24,8)(2,3,10,17,12,16,11,15)(4,19,20,26,21,25,32,22)(9,18,27,31,29,30,28,14)

Polycyclic presentation:

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

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, 380394);