#93934

order	512
nilpotency class	5
isoclinism	isoclinic class of #7422
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1347
commutator	C₂×C₁₆	C₂×C₈
Frattini, cosocle	C₂×#346	C₂²
prime radical	C₂³	#357
LCS	G ▷ C₂×C₁₆ ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

133 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 133 conjugacy classes of subgroups — too many to draw. Only its 50 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 2 generators on 40 points:

(2,5,6,7)(9,10,12,19,14,22,25,17,15,23,28,18,13,21,29,35,16,20,30,34,31,36,40,33,32,38,11,24,26,37,27,39);(1,2)(3,7)(4,6)(5,8)(10,17,20,33)(11,25)(12,27)(13,31)(14,26)(15,32)(18,21,24,37)(19,36,34,22)(23,39,38,35)(28,40)(29,30)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,f^2=hi,g^2=h,h^2=i,b^a=bc,c^a=ce,e^a=ei,c^b=cf,d^b=defi,f^b=fh,g^b=ghi,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1130);