#508069

order	512
nilpotency class	4
isoclinism	isoclinic class of #508066
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂²×D₄	C₂⁴
prime radical	#23902	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1129 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 1129 conjugacy classes of subgroups — too many to draw. Only its 124 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)(3,6)(4,5)(7,8)(9,10,14,18)(11,19,22,33)(12,23,25,16)(13,17,26,31)(15,27,29,21)(20,34,36,32)(24,38,39,30)(28,35,37,40);(2,5)(6,8)(13,26)(17,31)(24,39)(30,38);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,21)(14,22)(16,28)(17,19)(18,29)(23,37)(24,35)(25,36)(26,27)(30,32)(31,33)(34,38)(39,40);(3,7)(6,8)(9,12,14,25)(10,16,18,23)(11,20,22,36)(13,24,26,39)(15,28,29,37)(17,30,31,38)(19,32,33,34)(21,35,27,40)

Polycyclic presentation:

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

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415265);