#121714

order	512
nilpotency class	3
isoclinism	isoclinic class of #121711
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

656 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 126 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=deh,c^2=de,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,e^b=ei,f^b=fi,g^b=gh,d^c=di,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28910);