#413271

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

1108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=f,d^2=h,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,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, 320467);