#125065

order	512
nilpotency class	4
isoclinism	isoclinic class of #125064
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂×#466	C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

383 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 383 conjugacy classes of subgroups — too many to draw. Only its 80 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32261);