#134244

order	512
nilpotency class	4
isoclinism	isoclinic class of #11521
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#410
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1466
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

516 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 114 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=d,d^2=i,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,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, 41440);