#142309

order	512
nilpotency class	4
isoclinism	isoclinic class of #142308
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

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

normal subgroups only

This group has 578 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=gi,e^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,g^b=gi,h^b=hi,d^c=dh,f^c=fi,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250258, #72250333, #72250353
maximal normal subgroup of	#72250258, #72250333, #72250353
maximal quotient of	#72264968, #72307653

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49505);