#10534385

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	#45655	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4215 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=hi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10441581);