#407933

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

977 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2,4,3)(5,6,8,9)(7,10,11,12)(13,14,20,25)(15,26,30,41)(16,31,32,44)(17,33,35,43)(18,36,19,24)(21,37,42,29)(22,38,40,28)(23,34,39,27);(2,3)(13,15,18,29,20,30,19,37)(14,21,24,26,25,42,36,41)(16,17,28,27,32,35,38,34)(22,43,31,23,40,33,44,39);(1,3,4,2)(6,9)(10,12)(13,16,19,38,20,32,18,28)(14,22,36,44,25,40,24,31)(15,27,37,17,30,34,29,35)(21,39,41,33,42,23,26,43);(5,7,8,11)(6,10,9,12)(13,17,18,27,20,35,19,34)(14,23,24,33,25,39,36,43)(15,28,29,32,30,38,37,16)(21,40,26,44,42,22,41,31)

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=ei,d^2=f,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, 315129);