#10332171

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	#49589	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4252 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)(3,9,10,8)(4,5,13,14)(11,15,16,12)(17,18)(19,23)(20,28)(21,22)(24,29)(25,27)(26,31)(30,32)(33,34)(35,41)(36,38)(37,40)(39,43)(42,47)(44,45)(46,48);(2,7)(3,10)(5,14)(11,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32)(33,35,38,43)(34,39,36,41)(37,42,45,48)(40,46,44,47);(2,7)(4,13)(8,9)(11,16)(18,22)(23,25)(24,28)(30,31)(37,45)(40,44)(42,48)(46,47);(1,3,6,10)(2,8,7,9)(4,11,13,16)(5,12,14,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32)(33,36,38,34)(35,39,43,41)(37,40,45,44)(42,47,48,46);(1,4)(2,5)(3,11)(6,13)(7,14)(8,12)(9,15)(10,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48);(1,5)(2,4)(3,12)(6,14)(7,13)(8,11)(9,16)(10,15)(17,20)(18,24)(19,26)(21,29)(22,28)(23,30)(25,31)(27,32)(33,35)(34,41)(36,39)(37,42)(38,43)(40,47)(44,46)(45,48)

Polycyclic presentation:

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

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, 10239367);