#508094

order	512
nilpotency class	4
isoclinism	isoclinic class of #508070
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	#23908	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1100 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 1100 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)(13,25)(15,28)(17,26)(20,33)(21,27)(23,37)(29,39)(30,32)(31,36)(34,40)(35,38);(2,5)(6,8)(9,10,14,18)(11,15,21,30)(12,22,24,16)(13,26,25,17)(19,28,27,32)(20,34,35,29)(23,31,37,36)(33,40,38,39);(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)(22,34)(23,38)(24,35)(26,28)(31,40)(33,37)(36,39);(3,7)(6,8)(9,12,14,24)(10,16,18,22)(11,20,21,35)(13,23,25,37)(15,29,30,34)(17,31,26,36)(19,33,27,38)(28,39,32,40)

Polycyclic presentation:

2^9|b^2=i,d^2=i,f^2=i,b^a=be,c^a=cf,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, 415290);