#353015

order	512
nilpotency class	3
isoclinism	isoclinic class of #352064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

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

1009 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1009 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) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=hi,b^2=eh,c^2=fi,d^2=g,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=di,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, 260211);