#129351

order	512
nilpotency class	4
isoclinism	isoclinic class of #11502
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	C₂⁴	C₂²×C₈
Frattini, cosocle	C₂²×M₁₆	C₂³
socle	C₂²	#1459
prime radical	C₂×#525	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 850 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)(5,6,8,13,9,14,19,26,10,11,18,22,20,23,30,33)(7,12,16,24,17,25,29,34,15,21,27,31,28,32,35,36);(6,11)(7,15)(14,23)(16,27)(17,28)(24,31)(29,35)(34,36);(2,4)(5,7,9,17,10,15,20,28)(6,12,14,25,11,21,23,32)(8,16,19,29,18,27,30,35)(13,24,26,34,22,31,33,36)

Polycyclic presentation:

2^9|a^2=f,c^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,f^b=fg,f^d=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, 36547);