#129385

order	512
nilpotency class	3
isoclinism	isoclinic class of #8304
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

688 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 688 conjugacy classes of subgroups — too many to draw. Only its 142 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) = 256

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=gh,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,c^b=ci,f^b=fg

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36581);