#129808

order	512
nilpotency class	4
isoclinism	isoclinic class of #129741
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄⋊C₄	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

892 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 892 conjugacy classes of subgroups — too many to draw. Only its 108 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fghi,d^c=di,f^d=fi

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, 37004);