#238084

order	512
nilpotency class	3
isoclinism	isoclinic class of #238082
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#393
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1632 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 1632 conjugacy classes of subgroups — too many to draw. Only its 178 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)(3,5)(4,6)(7,8)(9,10)(11,15)(12,22,25,16)(13,26)(14,18)(17,27)(19,32,23,28)(20,34)(21,30)(24,31,38,36)(29,35)(33,39,37,40);(1,3)(2,5)(4,7)(6,8)(9,11,14,21)(10,15,18,30)(12,23)(13,20,27,35)(16,28)(17,29,26,34)(19,25)(22,32)(24,37)(31,39)(33,38)(36,40);(5,8)(9,12)(10,16)(11,19)(13,24)(14,25)(15,28)(17,31)(18,22)(20,33)(21,23)(26,36)(27,38)(29,39)(30,32)(34,40)(35,37);(9,13,14,27)(10,17,18,26)(11,20,21,35)(12,24,25,38)(15,29,30,34)(16,31,22,36)(19,33,23,37)(28,39,32,40)

Polycyclic presentation:

2^9|a^2=e,b^2=eh,d^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145280);