#133752

order	512
nilpotency class	4
isoclinism	isoclinic class of #133281
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	#36721	C₂
LCS	G ▷ C₂⁵ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

889 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 889 conjugacy classes of subgroups — too many to draw. Only its 80 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) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40948);