#512128

order	512
nilpotency class	4
isoclinism	isoclinic class of #511796
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×Q₈	C₂⁴
prime radical	#23909	C₂
LCS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1058 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 1058 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)(12,16)(13,17)(14,20)(18,24)(19,26)(21,28)(22,29)(23,32)(25,33)(27,34)(30,35)(31,37)(36,39)(38,40);(2,5)(6,8)(9,11,14,23)(10,15,20,32)(12,21,24,35)(13,22,26,37)(16,28,18,30)(17,29,19,31)(25,36,27,38)(33,39,34,40);(1,3,4,7)(2,6,5,8)(10,16)(12,24)(13,26)(15,28)(17,33)(18,20)(19,34)(21,35)(22,37)(29,39)(30,32)(31,40);(10,17)(13,26)(15,29)(16,33)(18,34)(19,20)(22,37)(25,27)(28,39)(30,40)(31,32)(36,38)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419324);