#110574

order	512
nilpotency class	3
isoclinism	isoclinic class of #9385
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

832 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 832 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gh,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239293
maximal normal subgroup of	#72239293
maximal quotient of	#72239294

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17770);