#92970

order	512
nilpotency class	4
isoclinism	isoclinic class of #353
automorphism	?
homology	Abelianization (H₁) = C₄×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#99
commutator	C₂×C₄	C₄×C₁₆
Frattini, cosocle	C₂×#431	C₂²
socle	C₂²	#1280
prime radical	C₂⁴	C₄⋊C₈
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×#431 ◁ G
LECS	G ▷ C₂×#431 ▷ C₂²×C₄ ▷ C₂² ▷ C₁

270 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 128

A minimal faithful permutation representation — 2 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=g,e^2=hi,g^2=i,b^a=bc,c^a=cf,e^a=eh,f^a=fh,d^b=dfh,d^c=dh

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 166);