#149592

order	512
nilpotency class	5
isoclinism	isoclinic class of #13603
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#354
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂	C₂×#1319
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 369 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) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56788);