#116880

order	512
nilpotency class	3
isoclinism	isoclinic class of #9974
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1439
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1458 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 1458 conjugacy classes of subgroups — too many to draw. Only its 134 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)(3,5)(4,6)(7,8)(9,10,14,18,15,19,13,29)(11,20,23,31,24,35,22,16)(12,25,27,17,28,32,26,33)(21,30,37,34,38,40,36,39);(1,3)(2,5)(4,7)(6,8)(9,11,13,22,15,24,14,23)(10,16,18,20,19,31,29,35)(12,21,26,36,28,38,27,37)(17,30,32,34,33,40,25,39);(5,8)(9,12)(10,17)(11,21)(13,26)(14,27)(15,28)(16,30)(18,32)(19,33)(20,34)(22,36)(23,37)(24,38)(25,29)(31,40)(35,39)

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,e^b=ei,e^c=eh

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, 24076);