#503363

order	512
nilpotency class	4
isoclinism	isoclinic class of #501120
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂³	C₂×#120
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

723 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 723 conjugacy classes of subgroups — too many to draw. Only its 148 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72271508
maximal normal subgroup of	#72271508
maximal quotient of	#72271531

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410559);