#122634

order	512
nilpotency class	4
isoclinism	isoclinic class of #122554
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#327
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	#23386	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

694 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 694 conjugacy classes of subgroups — too many to draw. Only its 76 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)(9,10,13,18,15,19,14,30)(11,20,23,34,21,37,24,40)(12,25,27,17,29,35,28,36)(16,26,32,38,31,39,33,22);(2,5)(6,7)(9,11)(10,16)(12,26)(13,24)(14,23)(15,21)(17,34)(18,32)(19,31)(20,25)(22,27)(28,38)(29,39)(30,33)(35,37)(36,40);(11,21)(16,31)(20,37)(22,38)(23,24)(26,39)(32,33)(34,40)

Polycyclic presentation:

2^9|a^2=e,d^2=gi,e^2=i,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,g^a=gi,c^b=ci,d^b=dg,e^b=efgi,f^b=fi,g^b=gi,e^d=ehi

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, 29830);