#9912894

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

3816 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)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,21,24)(19,22,25,29)(20,23,28,32)(26,30,27,31)(33,34,39,41)(35,42,36,37)(38,40,47,48)(43,46,44,45);(3,9)(7,13)(10,15)(14,16)(19,25)(22,29)(26,27)(30,31)(33,35,39,36)(34,37,41,42)(38,43,47,44)(40,45,48,46);(2,6)(7,13)(8,11)(14,16)(17,19)(18,22)(20,26)(21,25)(23,30)(24,29)(27,28)(31,32)(33,36,39,35)(34,37,41,42)(38,44,47,43)(40,45,48,46);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,19)(18,22)(20,27)(21,25)(23,31)(24,29)(26,28)(30,32)(33,37)(34,35)(36,41)(38,46)(39,42)(40,44)(43,48)(45,47);(1,4,5,12)(2,8,6,11)(3,10,9,15)(7,14,13,16)(17,20)(18,23)(19,26)(21,28)(22,30)(24,32)(25,27)(29,31)(33,38)(34,40)(35,43)(36,44)(37,45)(39,47)(41,48)(42,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,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, 9820090);