#407371

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

826 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=efh,c^2=f,d^2=ei,e^2=i,f^2=i,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,g^b=gi,d^c=di,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, 314567);