#10324604

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	#2613	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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)(3,4)(5,13)(6,7)(8,15)(9,12)(10,16)(11,14)(17,18,22,26)(19,23,27,32)(20,24,30,29)(21,25,31,28)(33,34,38,42)(35,43,45,39)(36,46,44,40)(37,41,47,48);(1,3,6,9)(2,4,7,12)(5,14,15,10)(8,16,13,11)(19,27)(21,31)(23,32)(25,28)(33,35)(34,39)(36,47)(37,44)(38,45)(40,48)(41,46)(42,43);(2,7)(3,9)(8,13)(10,14)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,36,38,44)(34,40,42,46)(35,37,45,47)(39,41,43,48);(1,4)(2,3)(5,11)(6,12)(7,9)(8,10)(13,14)(15,16)(17,19)(18,23)(20,25)(21,29)(22,27)(24,31)(26,32)(28,30)(33,35)(34,39)(36,37)(38,45)(40,41)(42,43)(44,47)(46,48);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,15)(14,16)(17,20)(18,24)(19,28)(21,23)(22,30)(25,27)(26,29)(31,32)(33,37)(34,41)(35,44)(36,45)(38,47)(39,46)(40,43)(42,48);(1,5)(2,8)(3,10)(4,11)(6,15)(7,13)(9,14)(12,16)(17,21)(18,25)(19,29)(20,23)(22,31)(24,27)(26,28)(30,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,45)(44,46)(47,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^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, 10231800);