#128328

order	512
nilpotency class	3
isoclinism	isoclinic class of #1462
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

510 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 510 conjugacy classes of subgroups — too many to draw. Only its 170 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 — 3 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,e^2=i,f^2=h,h^2=i,b^a=bd,f^a=fg,c^b=ci,d^b=dg

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35524);