#142416

order	512
nilpotency class	4
isoclinism	isoclinic class of #142224
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2021
commutator, Frattini, cosocle	C₂×#117	C₂³
prime radical	#23445	C₂
LCS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

788 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 70 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=g,e^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fhi,c^b=cf,d^b=dg,h^b=hi,d^c=dh,e^c=ei,f^d=fi

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, 49612);