#93157

order	512
nilpotency class	4
isoclinism	isoclinic class of #93155
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

592 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 592 conjugacy classes of subgroups — too many to draw. Only its 48 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)(11,14,17,21)(12,19,20,15)(13,16)(18,22)(23,24);(2,5,7,6)(9,11,12,18,13,17,20,23)(10,14,15,22,16,21,19,24)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,e^2=i,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=gi,c^b=cg,d^b=dfh,f^b=fi,g^b=gi,d^c=dh,e^c=ei,e^d=ei

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 353);