#153623

order	512
nilpotency class	6
isoclinism	isoclinic class of #153622
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13673
commutator, Frattini, cosocle	#494	C₂³
LCS	G ▷ #494 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×C₄ ◁ C₄² ◁ #494 ◁ G
LECS	G ▷ #494 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

276 conjugacy classes of subgroups; only the 30 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 30 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)(3,14)(4,17)(5,9)(6,12)(7,13)(8,15)(10,20)(11,21)(16,28)(18,26)(19,24)(22,29)(23,30)(25,32)(27,31);(1,3,7,6)(2,8,13,26)(4,18,20,15)(5,16,21,23)(9,27,11,24)(10,12,17,14)(19,32,31,22)(25,30,29,28);(2,9)(3,6)(4,15)(5,21)(8,24)(10,25)(11,13)(12,28)(14,30)(17,29)(18,20)(19,32)(22,31)(26,27)

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,f^a=fg,g^a=gi,h^a=hi,c^b=ci,d^b=di,e^b=efgh,g^b=gh,d^c=df,e^c=ei,f^c=fhi,g^c=ghi,e^d=egh,f^d=fhi,g^d=gi,f^e=fhi,g^e=gi

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283604, #72283696
maximal normal subgroup of	#72283604, #72283696

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60819);