#122983

order	512
nilpotency class	4
isoclinism	isoclinic class of #122971
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1595
commutator, Frattini, cosocle	C₂×C₂²⋊C₈	C₂³
prime radical	#23923	C₂
LCS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂×C₂²⋊C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₁

654 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 654 conjugacy classes of subgroups — too many to draw. Only its 44 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)(4,6)(7,8)(9,10,15,19)(11,20,23,38)(12,17,14,29)(13,25,28,39)(16,31,34,22)(18,24,36,27)(21,37,30,35)(26,32,40,33);(2,5)(6,8)(9,11)(10,16)(12,21)(13,26)(14,30)(15,23)(17,32)(18,35)(19,34)(20,25)(22,27)(24,31)(28,40)(29,33)(36,37)(38,39);(3,7)(9,12,15,14)(10,17,19,29)(11,21,23,30)(13,24,28,27)(16,32,34,33)(18,25,36,39)(20,37,38,35)(22,26,31,40)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=fh,f^2=h,b^a=bd,c^a=ce,d^a=df,f^a=fh,g^a=gi,d^b=df,e^b=egi,f^b=fh,d^c=dg,e^d=ei

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 30179);