#102700

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

335 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=g,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, 9896);