#101512

order	512
nilpotency class	3
isoclinism	isoclinic class of #101351
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

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

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

normal subgroups only

This group has 341 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) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=g,d^2=gh,e^2=h,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=ehi,c^b=cg,d^b=dh,e^b=ei,f^b=fh,d^c=di,e^c=eh,f^c=fi

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, 8708);