#135572

order	512
nilpotency class	4
isoclinism	isoclinic class of #135522
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

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

normal subgroups only

This group has 623 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:

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

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,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	#72234503, #72234508
maximal normal subgroup of	#72234503, #72234508

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42768);