#110132

order	512
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

692 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 692 conjugacy classes of subgroups — too many to draw. Only its 162 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 — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,d^2=i,e^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,f^c=fh

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, 17328);