#10023013

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	#41158	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4003 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^c=egi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9930209);