#102703

order	512
nilpotency class	3
isoclinism	isoclinic class of #102685
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
prime radical	C₂×#127	D₄
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

402 conjugacy classes of subgroups; only the 64 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 64 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=g,e^2=i,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=ei,f^a=fghi,c^b=cf,d^b=dg,e^b=egi,f^b=fgh,d^c=dh,e^c=ei,f^c=fgh

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9899);