#418117

order	512
nilpotency class	3
isoclinism	isoclinic class of #417460
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1714 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 1714 conjugacy classes of subgroups — too many to draw. Only its 226 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^c=fi,h^c=hi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325313);