#103239

order	512
nilpotency class	3
isoclinism	isoclinic class of #103237
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#32902	C₂
LCS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

1044 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 1044 conjugacy classes of subgroups — too many to draw. Only its 96 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) = 128

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

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

Polycyclic presentation:

2^9|c^2=i,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fh,c^b=cf,d^b=dg,d^c=dh,e^c=ei

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10435);