#10412468

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	#2846	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

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, 10319664);