#102708

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, prime radical	C₂³	#392
commutator, Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₄³ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

359 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=gi,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, 9904);