#121706

order	512
nilpotency class	3
isoclinism	isoclinic class of #121705
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	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 636 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)(11,16,24,32,22,33,23,31)(12,25,27,35,29,36,28,17)(21,34,38,39,37,40,26,30);(1,3)(2,5)(4,8)(6,7)(9,11,13,24,15,22,14,23)(10,16,20,31,18,33,19,32)(12,26,27,37,29,38,28,21)(17,34,36,39,35,40,25,30);(1,3)(2,5)(4,8)(6,7)(9,12,13,27,15,29,14,28)(10,17,20,36,18,35,19,25)(11,21,24,26,22,37,23,38)(16,30,31,34,33,39,32,40)

Polycyclic presentation:

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

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, 28902);