#10442560

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	#2416	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=hi,e^2=gi,f^2=h,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, 10349756);