#454534

order	512
nilpotency class	3
isoclinism	isoclinic class of #453952
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1356 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1356 conjugacy classes of subgroups — too many to draw. Only its 218 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267777
maximal normal subgroup of	#72267777
maximal quotient of	#72267778

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 361730);