#142311

order	512
nilpotency class	4
isoclinism	isoclinic class of #142310
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#575	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

570 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

(1,2)(3,4)(5,6,12,16)(7,17,22,35)(8,11,15,10)(9,25,28,24)(13,30,18,19)(14,29,33,27)(20,32,21,34)(23,26,31,36);(5,7)(6,13)(8,23)(9,26)(10,20)(11,21)(12,22)(14,32)(15,31)(16,18)(17,25)(19,27)(24,35)(28,36)(29,30)(33,34);(2,4)(5,8,10,6,12,15,11,16)(7,18,21,31,22,13,20,23)(9,24,27,14,28,25,29,33)(17,26,32,30,35,36,34,19)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,g^a=gi,c^b=cf,d^b=dg,f^b=fi,g^b=gi,h^b=hi,d^c=dh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250331, #72250333, #72250369
maximal normal subgroup of	#72250331, #72250333, #72250369

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49507);