#352990

order	512
nilpotency class	3
isoclinism	isoclinic class of #352174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

910 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 910 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=e,c^2=f,d^2=gh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,c^b=ci,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260186);