#512165

order	512
nilpotency class	4
isoclinism	isoclinic class of #511741
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

690 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 174 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=e,d^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gh,h^a=hi,g^d=gh,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419361);