#151252

order	512
nilpotency class	5
isoclinism	isoclinic class of #151252
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12141
commutator, Frattini, cosocle	#423	C₂³
prime radical	#12173	C₂
LCS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #423 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

230 conjugacy classes of subgroups; only the 36 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 36 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:

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

Polycyclic presentation:

2^9|c^2=fh,d^2=fi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fi,g^a=gh,d^b=df,e^b=eghi,f^b=fi,h^b=hi,d^c=dg,e^c=ehi,g^c=gi,e^d=eh,g^d=gi

Conjugacy classes & character table

29 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58448);