#134294

order	512
nilpotency class	4
isoclinism	isoclinic class of #11505
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

518 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 110 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dh,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,e^c=ei,f^c=fgi,f^e=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, 41490);