#401468

order	512
nilpotency class	3
isoclinism	isoclinic class of #395240
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1097 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=ef,d^2=ef,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,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308664);