#100801

order	512
nilpotency class	3
isoclinism	isoclinic class of #100715
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

448 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 448 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 48 points:

(2,6)(3,4)(7,8)(9,10,14,16,13,11,12,15)(17,18)(19,23,28,38)(20,30)(21,33)(22,25)(24,31)(26,35)(27,41,42,48)(29,45,44,37)(32,40)(34,39)(36,46,47,43);(1,2,3,7,5,8,4,6)(9,11,13,10)(12,15,14,16)(17,19)(18,23,25,38)(20,27)(21,29)(22,28)(24,36,39,47)(26,37,33,45)(30,41,40,48)(31,43)(32,42)(34,46)(35,44);(1,3,5,4)(2,7,8,6)(12,14)(15,16)(17,20,22,32)(18,24,25,39)(19,27,28,42)(21,34,35,31)(23,36,38,47)(26,40,33,30)(29,46,44,43)(37,48,45,41)

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=g,d^2=gh,e^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=ehi,f^c=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7997);