#8893810

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8801006);