#512315

order	512
nilpotency class	4
isoclinism	isoclinic class of #511963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2218	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

570 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=efhi,d^2=e,e^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,h^a=hi,c^b=ci,g^d=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419511);