#150645

order	512
nilpotency class	5
isoclinism	isoclinic class of #150564
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×#353
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#391	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₄² ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ #2297 ◁ G
LECS	G ▷ #391 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

391 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 391 conjugacy classes of subgroups — too many to draw. Only its 56 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=i,d^2=ghi,e^2=i,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dg,e^b=efgh,e^d=ehi,f^e=fi

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, 57841);