#135574

order	512
nilpotency class	4
isoclinism	isoclinic class of #135524
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 583 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42770);