#134293

order	512
nilpotency class	4
isoclinism	isoclinic class of #11502
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	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1459
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 520 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)(5,6,8,13,9,14,19,26,10,12,18,24,20,25,30,34)(7,11,16,22,17,23,29,33,15,21,27,31,28,32,35,36);(1,2,3,4)(5,7,9,17,10,15,20,28)(6,11,14,23,12,21,25,32)(8,16,19,29,18,27,30,35)(13,22,26,33,24,31,34,36);(6,12)(7,15)(14,25)(16,27)(17,28)(22,31)(29,35)(33,36)

Polycyclic presentation:

2^9|a^2=f,b^2=dh,f^2=h,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=ci,f^c=fg,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41489);