#135520

order	512
nilpotency class	4
isoclinism	isoclinic class of #135518
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1819
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

646 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 646 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72260536, #72260645
maximal normal subgroup of	#72260536, #72260645

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42716);