#9912915

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3812 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820111);