#129263

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	C₂×#324
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

884 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 884 conjugacy classes of subgroups — too many to draw. Only its 138 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)(5,6,8,13,9,14,19,27,10,12,18,24,20,25,31,34)(7,11,16,21,17,23,30,32,15,22,28,26,29,33,36,35);(5,7)(6,11,12,22)(8,16)(9,17)(10,15)(13,26,24,21)(14,23,25,33)(18,28)(19,30)(20,29)(27,35,34,32)(31,36);(2,4)

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,f^b=fg,f^d=fi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308032, #72308119
maximal normal subgroup of	#72308032, #72308119

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36459);