#92983

order	512
nilpotency class	4
isoclinism	isoclinic class of #7307
automorphism	?
homology	Abelianization (H₁) = C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#342
commutator	C₂²×C₄	C₄×C₈
Frattini, cosocle	C₂²×C₄⋊C₈	C₂²
socle	C₂²	#1280
prime radical	C₂⁴	C₂²⋊C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂²×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

310 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 64

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

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

Polycyclic presentation:

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

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, 179);