#137751

order	512
nilpotency class	4
isoclinism	isoclinic class of #11541
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1459
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

545 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 545 conjugacy classes of subgroups — too many to draw. Only its 104 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dgi,e^b=ei,f^b=fg,g^b=gi,d^c=di,f^c=fi,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44947);