#10328106

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	#53471	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3940 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)(3,10)(4,5)(6,7)(8,11)(9,14)(12,16)(13,15)(17,18)(19,24)(20,26)(21,27)(22,29)(23,25)(28,32)(30,31)(33,34,38,42)(35,39,43,48)(36,40,46,45)(37,41,47,44);(2,7)(3,11)(5,14)(12,15)(17,19,23,21)(18,24,25,27)(20,28,31,29)(22,26,32,30)(35,43)(37,47)(39,48)(41,44);(2,7)(4,9)(8,10)(12,15)(18,25)(19,21)(26,30)(28,29)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,3,6,11)(2,8,7,10)(4,12,9,15)(5,13,14,16)(17,20,23,31)(18,26,25,30)(19,29,21,28)(22,27,32,24)(33,35)(34,39)(36,41)(37,45)(38,43)(40,47)(42,48)(44,46);(1,4,6,9)(2,5,7,14)(3,12,11,15)(8,13,10,16)(17,21,23,19)(18,27,25,24)(20,28,31,29)(22,26,32,30)(33,36)(34,40)(35,44)(37,39)(38,46)(41,43)(42,45)(47,48);(1,5)(2,9)(3,13)(4,7)(6,14)(8,15)(10,12)(11,16)(17,22,23,32)(18,28,25,29)(19,30,21,26)(20,27,31,24)(33,37)(34,41)(35,45)(36,39)(38,47)(40,43)(42,44)(46,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=h,e^2=h,f^2=gh,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, 10235302);