#9912935

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	#2555	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3799 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)(3,7)(4,8)(5,6)(9,14)(10,13)(11,12)(15,16)(17,18,21,24)(19,22,25,29)(20,23,28,32)(26,30,27,31)(33,34,40,37)(35,43,42,39)(36,46,38,41)(44,47,45,48);(2,6)(7,13)(8,11)(14,15)(19,25)(22,29)(26,27)(30,31)(33,35)(34,39)(36,44)(37,43)(38,45)(40,42)(41,47)(46,48);(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,28)(31,32)(33,36,40,38)(34,41,37,46)(35,44,42,45)(39,47,43,48);(1,3,5,10)(2,7,6,13)(4,9,12,16)(8,14,11,15)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32)(33,37,40,34)(35,43,42,39)(36,41,38,46)(44,47,45,48);(1,2)(3,7)(4,11)(5,6)(8,12)(9,15)(10,13)(14,16)(17,19)(18,22)(20,27)(21,25)(23,31)(24,29)(26,28)(30,32)(33,38,40,36)(34,41,37,46)(35,45,42,44)(39,47,43,48);(1,4)(2,8)(3,9)(5,12)(6,11)(7,14)(10,16)(13,15)(17,20)(18,23)(19,26)(21,28)(22,30)(24,32)(25,27)(29,31)(33,39,40,43)(34,42,37,35)(36,47,38,48)(41,45,46,44)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^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, 9820131);