#128385

order	512
nilpotency class	3
isoclinism	isoclinic class of #105580
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₄×C₂²⋊C₄
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1749
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

450 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=eg,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fgi,c^b=ch,d^b=dg,e^b=ei,f^c=fi

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, 35581);