#9935922

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	#51023	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4156 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,20)(21,24)(22,30)(23,25)(26,31)(27,32)(28,29)(33,34,39,41)(35,40,42,38)(36,45,47,37)(43,44,46,48);(3,9)(7,13)(10,15)(14,16)(17,19,23,21)(18,24,25,20)(22,27,31,29)(26,32,30,28)(33,35)(34,40)(36,46)(37,44)(38,41)(39,42)(43,47)(45,48);(2,6)(7,13)(8,11)(14,16)(18,25)(20,24)(26,30)(28,32)(34,41)(35,42)(36,47)(44,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20,23,24)(18,19,25,21)(22,28,31,32)(26,27,30,29)(33,36)(34,37)(35,43)(38,44)(39,47)(40,48)(41,45)(42,46);(1,3)(2,7)(4,10)(5,9)(6,13)(8,14)(11,16)(12,15)(17,21,23,19)(18,24,25,20)(22,27,31,29)(26,28,30,32)(33,37)(34,36)(35,44)(38,43)(39,45)(40,46)(41,47)(42,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22,23,31)(18,26,25,30)(19,27,21,29)(20,28,24,32)(33,38)(34,42)(35,41)(36,44)(37,46)(39,40)(43,45)(47,48)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,d^c=dh,f^e=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, 9843118);