#244708

order	512
nilpotency class	3
isoclinism	isoclinic class of #244655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1365 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1365 conjugacy classes of subgroups — too many to draw. Only its 154 normal subgroups are shown.

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

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 16 points:

(2,6)(3,7)(4,8)(9,10,14,11)(12,15,13,16);(1,2,5,3)(4,7,8,6)(9,11,14,10)(12,15,13,16);(4,8)(6,7)(10,15)(11,16)(12,13);(1,3,5,2)(4,6,8,7)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=h,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,e^b=eh,f^b=fh,g^b=gi,e^c=eh,f^c=fi,g^c=gi,e^d=eh,f^d=fh

maximal subgroup of	#72226259
maximal normal subgroup of	#72226259

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151904);