#120616

order	512
nilpotency class	3
isoclinism	isoclinic class of #10260
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#121
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂³	#389
prime radical	C₂⁴	#125
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

456 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

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

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2,4,3)(5,6,9,14,10,15,17,13)(7,16,8,11,18,12,19,20)(21,22,26,29)(23,30,35,36)(24,31,28,25)(27,34,32,33);(1,3,4,2)(5,7,17,19,10,18,9,8)(6,11,14,12,15,20,13,16)(21,23,31,34,26,35,25,33)(22,27,24,36,29,32,28,30);(2,3)(5,8,9,18,10,19,17,7)(6,12,13,11,15,16,14,20)(21,24,25,22,26,28,31,29)(23,32,33,36,35,27,34,30)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=d,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,f^b=fi,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 27812);