#142393

order	512
nilpotency class	4
isoclinism	isoclinic class of #142392
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	C₂⁵	SD₁₆
LCS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 502 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)(5,6,10,13)(7,15,18,34)(8,19,22,36)(9,24,21,14)(11,26,27,16)(12,29,31,23)(17,30,25,33)(20,28,35,32);(5,7,9,25,10,18,21,17)(6,11,14,32,13,27,24,28)(8,20,23,26,22,35,29,16)(12,30,19,34,31,33,36,15);(2,4)(11,27)(16,26)(20,35)(28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gi,g^2=i,b^a=bd,c^a=ce,d^a=dg,f^a=fhi,g^a=gi,c^b=cf,d^b=dgi,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	#72250353, #72250363, #72250365
maximal normal subgroup of	#72250353, #72250363, #72250365

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 49589);