#122278

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
socle	C₂²	C₂×#323
prime radical	C₂²×#41	C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

439 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 86 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)(3,7)(4,8,12,6)(5,9)(10,14,16,13)(11,15)(17,18,20,25,21,26,29,32,22,23,28,30,19,24,27,31);(2,6)(4,12)(7,13)(8,9)(10,16)(14,15)(18,23)(24,26);(1,3,5,11)(2,7,9,15)(4,10,12,16)(6,13,8,14)(17,19,22,21)(18,24,23,26)(20,27,28,29)(25,31,30,32)

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=hi,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,d^b=dhi,e^b=efhi,e^d=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29474);