#138078

order	512
nilpotency class	4
isoclinism	isoclinic class of #138078
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#9118
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2873 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 375 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) = 32

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=g,d^2=gh,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dg,h^a=hi,c^b=cg,d^b=dh,e^b=ei,f^b=fhi,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 45274);