#409633

order	512
nilpotency class	3
isoclinism	isoclinic class of #401790
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2485
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15043
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

799 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=f,g^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^b=gi,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, 316829);