#94445

order	512
nilpotency class	5
isoclinism	isoclinic class of #94445
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#7310
commutator	C₂²×Q₈	C₄²
Frattini, prime radical, cosocle	#3486	C₂²
LCS	G ▷ C₂²×Q₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂²×C₄ ◁ #3486 ◁ G
LECS	G ▷ #3486 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

368 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 368 conjugacy classes of subgroups — too many to draw. Only its 38 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 — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=i,f^2=i,b^a=bc,c^a=cf,e^a=egi,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,d^b=df,f^b=fh,d^c=dhi,f^c=fi,e^d=ehi,f^d=fi,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal quotient of

#72266409

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1641);