#508102

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	#23913	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1136 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 1136 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)(11,19)(12,22)(13,25)(15,28)(16,23)(17,26)(20,34)(21,27)(29,40)(30,32)(33,36)(35,39);(2,5)(6,8)(9,10,14,18)(11,15,21,30)(12,23,22,16)(13,26,25,17)(19,28,27,32)(20,35,36,29)(24,31,37,38)(33,40,34,39);(1,3)(2,6)(4,7)(5,8)(9,11)(10,15)(12,20)(13,27)(14,21)(16,29)(17,32)(18,30)(19,25)(22,36)(23,35)(24,34)(26,28)(31,40)(33,37)(38,39);(3,7)(6,8)(9,12,14,22)(10,16,18,23)(11,20,21,36)(13,24,25,37)(15,29,30,35)(17,31,26,38)(19,33,27,34)(28,39,32,40)

Polycyclic presentation:

2^9|b^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,g^a=ghi,c^b=cg,d^b=di,f^b=fh,d^c=de,f^c=fi,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, 415298);