#165926

order	512
nilpotency class	3
isoclinism	isoclinic class of #165873
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#2682	C₂
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

1271 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1271 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,e^2=g,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fi,e^b=eg,f^b=fh,e^c=eh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257810, #72257859, #72257862
maximal normal subgroup of	#72257810, #72257859, #72257862
maximal quotient of	#72265444, #72307573

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 73122);