#153483

order	512
nilpotency class	6
isoclinism	isoclinic class of #7523
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1404
commutator	#125	C₂²×C₄
Frattini, cosocle	#490	C₂³
socle	C₂	C₂×#1404
prime radical	#18990	C₂
LCS	G ▷ #125 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #3135 ◁ G
LECS	G ▷ #490 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

292 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 48 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)(3,8)(4,14,16,7)(5,10)(6,9,17,24)(11,25,27,19)(12,23)(13,22,28,32)(15,20,18,21)(26,30,29,31);(2,7)(4,15)(6,17)(8,19)(9,20)(10,21)(11,26)(13,28)(14,24)(22,30)(23,31)(25,32);(1,3,5,12)(2,8,10,23)(4,11,15,26)(6,13,17,28)(7,19,21,31)(9,22,24,32)(14,25,20,30)(16,27,18,29)

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=h,f^2=hi,g^2=i,b^a=bd,d^a=df,f^a=fg,g^a=gh,h^a=hi,d^b=dh,e^b=efi,f^b=fhi,g^b=gh,e^d=egh,f^d=fh,g^d=gi,f^e=fhi,g^e=gi

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, 60679);