#93132

order	512
nilpotency class	4
isoclinism	isoclinic class of #93129
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, prime radical, cosocle	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#121 ◁ G
LECS	G ▷ C₂²×#121 ▷ C₂⁴ ▷ C₂² ▷ C₁

599 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 599 conjugacy classes of subgroups — too many to draw. Only its 46 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=hi,d^2=i,b^a=bc,c^a=cf,e^a=eghi,f^a=fh,g^a=gi,c^b=cg,d^b=dfhi,f^b=fi,d^c=dh,e^d=ei

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328);