#8621975

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#55752	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4195 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8529171);