#410678

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1020 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1020 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,12)(8,11)(9,10)(13,14,18,26,20,24,19,25)(15,21,31,36,29,38,30,37)(16,23,17,22,33,40,32,39)(27,35,42,44,41,43,28,34);(2,4)(6,10)(11,12)(13,15,19,31,20,29,18,30)(14,21,26,37,24,38,25,36)(16,27,32,42,33,41,17,28)(22,34,40,43,39,44,23,35);(1,2,3,4)(5,7,9,8)(6,11,10,12)(13,16,18,17,20,33,19,32)(14,22,25,23,24,39,26,40)(15,27,30,28,29,41,31,42)(21,34,36,35,38,44,37,43);(5,8,9,7)(6,11,10,12)(13,17,18,33,20,32,19,16)(14,23,25,39,24,40,26,22)(15,28,30,41,29,42,31,27)(21,35,36,44,38,43,37,34)

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=fg,d^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,d^c=di

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, 317874);