#152925

order	512
nilpotency class	5
isoclinism	isoclinic class of #152923
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13221
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	#532	C₂³
prime radical	#24287	C₂
LCS	G ▷ C₂×#41 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #532 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

449 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 449 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=e,d^2=i,e^2=i,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gh,d^b=dg,f^b=fg,g^b=gh,h^b=hi,f^d=fhi,g^d=gi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60121);