#508073

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	#3485	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1070 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 1070 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)(9,10,15,11)(12,22,25,33)(13,20,27,17)(14,18,28,21)(16,29,19,24)(23,36,38,35)(26,34,39,31)(30,37,32,40);(2,5)(6,8)(9,11,15,10)(12,19,25,16)(13,17,27,20)(14,18,28,21)(22,24,33,29)(23,30,38,32)(26,34,39,31)(35,40,36,37);(1,3)(2,6)(4,7)(5,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,25)(17,30)(18,22)(20,32)(21,33)(26,37)(27,38)(28,29)(31,35)(34,36)(39,40);(3,7)(6,8)(9,13,15,27)(10,17,11,20)(12,23,25,38)(14,26,28,39)(16,30,19,32)(18,31,21,34)(22,35,33,36)(24,37,29,40)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=i,b^a=be,c^a=cf,d^a=di,g^a=ghi,c^b=cg,d^b=di,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, 415269);