#122414

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×C₄⋊C₈	C₂³
socle	C₂²	#1459
prime radical	#2397	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂×C₄⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

407 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 407 conjugacy classes of subgroups — too many to draw. Only its 76 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 — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=g,d^2=hi,e^2=g,g^2=i,b^a=bd,d^a=df,f^a=fh,d^b=dhi,e^b=efhi,e^d=eh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29610);