#93297

order	512
nilpotency class	4
isoclinism	isoclinic class of #93294
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1276
commutator	C₄²	C₄×C₈
Frattini, cosocle	C₂×C₄×M₁₆	C₂²
socle	C₂	#7017
prime radical	C₂⁴	#101
LCS	G ▷ C₄² ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×#97 ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×M₁₆ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

289 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 289 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fg,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=efg,f^a=fi,d^b=dg,h^b=hi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 493);