#94333

order	512
nilpotency class	5
isoclinism	isoclinic class of #7414
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1345
commutator	C₂×M₁₆	C₄²
Frattini, cosocle	C₂×#423	C₂²
socle	C₂	#7251
prime radical	C₂⁴	#104
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ C₂×#423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

273 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 273 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fi,d^2=i,e^2=h,f^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,h^a=hi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,g^c=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1529);