#142903

order	512
nilpotency class	4
isoclinism	isoclinic class of #142394
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1613
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

441 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 441 conjugacy classes of subgroups — too many to draw. Only its 72 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)(5,6,12,17)(7,18,22,31)(8,23,9,26)(10,16,11,15)(13,29,33,20)(14,24,35,25)(19,32,27,36)(21,34,28,30);(5,7,11,21,12,22,10,28)(6,13,16,32,17,33,15,36)(8,19,24,20,9,27,25,29)(14,34,26,18,35,30,23,31);(2,4)(7,19)(8,24)(9,25)(10,11)(13,30)(14,26)(15,16)(18,32)(20,28)(21,29)(22,27)(23,35)(31,36)(33,34)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gi,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fh,g^a=gi,h^a=hi,c^b=cf,d^b=dgi,f^b=fg,g^b=gi,d^c=dh,f^c=fgi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250528, #72250581, #72250587
maximal normal subgroup of	#72250528, #72250581, #72250587

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50099);