#134000

order	512
nilpotency class	4
isoclinism	isoclinic class of #133908
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2297 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

608 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 78 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)(3,6)(4,11,12,7)(5,8)(9,15,16,13)(10,14)(17,18)(19,24,27,22)(20,23)(21,26)(25,29)(28,30,32,31);(1,3,5,10)(2,6,8,14)(4,9,12,16)(7,13,11,15)(18,22,26,24)(20,29)(23,30,25,31)(28,32);(1,4)(2,7)(3,9)(5,12)(6,13)(8,11)(10,16)(14,15)(18,23,26,25)(19,27)(22,30,24,31)(28,32)

Polycyclic presentation:

2^9|a^2=f,b^2=gh,c^2=gi,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,f^b=fgi,d^c=di,f^c=fhi,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, 41196);