#512147

order	512
nilpotency class	4
isoclinism	isoclinic class of #511775
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×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

1114 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1114 conjugacy classes of subgroups — too many to draw. Only its 140 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)(11,15)(12,22)(13,17)(14,18)(16,24)(19,31)(20,27)(21,28)(23,35)(25,30)(26,33)(29,36)(32,39)(34,38)(37,40);(2,5)(6,8)(9,11,14,21)(10,15,18,28)(12,19,23,32)(13,20,25,34)(16,26,29,37)(17,27,30,38)(22,31,35,39)(24,33,36,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,24)(14,23)(15,26)(17,22)(18,29)(20,33)(21,32)(25,36)(27,31)(28,37)(30,35)(34,40)(38,39);(12,23)(13,25)(16,29)(19,32)(20,34)(22,35)(26,37)(31,39)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281265, #72281983, #72282650
maximal normal subgroup of	#72281265, #72281983, #72282650

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419343);