#151200

order	512
nilpotency class	5
isoclinism	isoclinic class of #151200
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

	subgroup	G/H
center, socle	C₂	#12124
commutator, Frattini, cosocle	#407	C₂³
prime radical	#10326	C₂
LCS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #407 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

407 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 16

A minimal faithful permutation representation — 3 generators on 16 points:

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

Polycyclic presentation:

2^9|b^2=fh,d^2=fi,f^2=i,b^a=bd,c^a=ce,d^a=df,f^a=fi,g^a=gh,c^b=ch,d^b=dfi,e^b=egh,f^b=fi,g^b=gi,h^b=hi,d^c=dg,f^c=fi,e^d=ehi,g^d=gi

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58396);