#123255

order	512
nilpotency class	4
isoclinism	isoclinic class of #123243
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	#3186	C₂²
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

287 conjugacy classes of subgroups; only the 42 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 128

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

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

Polycyclic presentation:

2^9|b^2=hi,c^2=g,d^2=fh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^c=ehi,e^d=ei

Conjugacy classes & character table

38 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30451);