#135523

order	512
nilpotency class	4
isoclinism	isoclinic class of #135522
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1819
commutator	C₂⁵	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂³ ▷ C₂ ▷ C₁

656 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 32

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=dh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,f^b=fg,e^c=egi,f^c=fh,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235148, #72235155, #72235171
maximal normal subgroup of	#72235148, #72235155, #72235171

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42719);