#124175

order	512
nilpotency class	4
isoclinism	isoclinic class of #1343
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#99
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	C₂×#97	C₂³
socle	C₂²	#1459
prime radical	#392	C₈
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

412 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 100 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)(3,5)(4,7)(6,8)(9,10,12,17,13,18,23,30,14,15,22,26,24,27,34,37)(11,16,19,28,20,29,32,38,21,25,31,35,33,36,39,40);(2,5)(3,8)(6,7)(10,15)(16,25)(18,27)(29,36);(9,11,13,20,14,21,24,33)(10,16,18,29,15,25,27,36)(12,19,23,32,22,31,34,39)(17,28,30,38,26,35,37,40)

Polycyclic presentation:

2^9|a^2=e,c^2=h,d^2=g,e^2=h,h^2=i,b^a=bd,d^a=df,f^a=fi,d^b=dg,e^b=efg,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31371);