#129350

order	512
nilpotency class	4
isoclinism	isoclinic class of #11499
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

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

normal subgroups only

This group has 894 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)(7,12,15,24,16,25,28,34,17,21,27,31,29,32,35,36);(6,11)(12,21)(14,23)(25,32);(2,4)(5,7,9,16,10,17,20,29)(6,12,14,25,11,21,23,32)(8,15,19,28,18,27,30,35)(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,f^b=fg,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308032
maximal normal subgroup of	#72308032

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36546);