#476094

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1346 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1346 conjugacy classes of subgroups — too many to draw. Only its 178 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,g^b=gi,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383290);