#10128208

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	#59384	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4188 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,34,38,36)(35,39,43,41)(37,44,45,40)(42,47,48,46);(2,7)(4,15)(8,11)(10,13)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(2,7)(3,12)(9,14)(10,13)(18,22)(23,25)(24,28)(31,32)(34,36)(39,41)(40,44)(46,47);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,4)(2,9)(3,13)(5,11)(6,15)(7,14)(8,16)(10,12)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48)

Polycyclic presentation:

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

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, 10035404);