#121734

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1445
prime radical	C₂⁴	C₄⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

498 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=i,f^2=hi,g^2=hi,b^a=bd,c^a=ce,f^a=fhi,c^b=cf,d^b=dh,e^b=eh,f^b=fhi,g^b=gh,d^c=di,e^c=ei,f^c=fhi,g^c=gi

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, 28930);