#499685

order	512
nilpotency class	4
isoclinism	isoclinic class of #499266
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

524 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,d^2=efh,g^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=dh,g^c=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265562, #72265599, #72265615, …
maximal normal subgroup of	#72265562, #72265599, #72265615, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406881);