#153018

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

252 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 44
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 44 points:

(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14)(15,27)(16,32)(17,37)(18,39)(19,41)(20,26)(21,30)(22,34)(23,28)(24,38)(25,40)(29,33)(31,35)(36,44)(42,43);(2,4)(6,9)(10,11)(13,15,28,26,18,31,36,29,40,43,42,25,33,44,35,39,20,23,27,14,21,37,16,24,19,22,34,41,38,32,17,30);(5,7,8,12)(6,10,9,11)(13,16,33,28,19,35,18,34,20,36,38,27,40,17,21,42)(14,22,30,31,25,23,24,32,26,43,39,37,41,15,29,44)

Polycyclic presentation:

2^9|a^2=f,b^2=def,c^2=ehi,d^2=gh,e^2=hi,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gh,h^a=hi,c^b=ci

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322759, #72322863, #72322888
maximal normal subgroup of	#72322759, #72322863, #72322888
maximal quotient of	#72322766

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60214);