#128357

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	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 580 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)(3,7)(4,6)(5,8)(9,10,14,12,16,15,13,18)(11,19,21,17,23,24,20,22);(1,3)(2,5,6,7)(4,8)(9,11,15,22,14,20,18,24,16,23,10,17,13,21,12,19);(9,12,14,15,16,18,13,10)(11,17,21,24,23,22,20,19)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,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, 35553);