#129355

order	512
nilpotency class	4
isoclinism	isoclinic class of #11505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1459
prime radical	C₂×#532	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

856 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 856 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,f^b=fgi,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36551);