#10023181

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	#56630	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4145 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,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, 9930377);