#121713

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,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, 28909);