#326615

order	512
nilpotency class	3
isoclinism	isoclinic class of #251800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2598
prime radical	#10032	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

560 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 126 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=e,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,d^c=dh,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233811);