#142805

order	512
nilpotency class	4
isoclinism	isoclinic class of #142256
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#2190	C₂
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 667 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)(5,6,12,18)(7,19,24,30)(8,25,27,15)(9,28,26,14)(10,17,11,16)(13,20,33,21)(22,34,35,32)(23,36,29,31);(5,7)(6,13)(8,22)(9,20)(10,29)(11,23)(12,24)(14,31)(15,30)(16,34)(17,32)(18,33)(19,25)(21,26)(27,35)(28,36);(2,4)(7,20)(8,9)(10,11)(13,30)(14,15)(16,17)(19,33)(21,24)(22,23)(25,28)(26,27)(29,35)(31,32)(34,36)

Polycyclic presentation:

2^9|a^2=i,d^2=gi,f^2=gi,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=dg,f^b=fg,g^b=gi,d^c=dh,f^c=fg,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	#72250264, #72250587, #72250615
maximal normal subgroup of	#72250264, #72250587, #72250615
maximal quotient of	#72307720

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 50001);