#9912918

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	#45962	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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)(17,18,23,24)(19,25,20,21)(22,26,31,32)(27,30,28,29)(33,34,37,40)(35,38,41,45)(36,39,44,48)(42,46,43,47);(1,3,7,12)(2,8,9,16)(4,13,15,5)(6,14,10,11)(18,24)(21,25)(26,32)(29,30)(35,41)(38,45)(42,43)(46,47);(2,9)(3,12)(4,15)(11,14)(17,19,23,20)(18,25,24,21)(22,27,31,28)(26,30,32,29)(33,35)(34,38)(36,42)(37,41)(39,46)(40,45)(43,44)(47,48);(1,4)(2,10)(3,13)(5,12)(6,9)(7,15)(8,11)(14,16)(17,20,23,19)(18,21,24,25)(22,28,31,27)(26,29,32,30)(33,34)(35,38)(36,39)(37,40)(41,45)(42,46)(43,47)(44,48);(1,5,7,13)(2,11,9,14)(3,4,12,15)(6,16,10,8)(17,21)(18,19)(20,24)(22,30)(23,25)(26,28)(27,32)(29,31)(33,35)(34,38)(36,43)(37,41)(39,47)(40,45)(42,44)(46,48);(1,6,7,10)(2,4,9,15)(3,14,12,11)(5,8,13,16)(17,22)(18,26)(19,27)(20,28)(21,29)(23,31)(24,32)(25,30)(33,36)(34,39)(35,42)(37,44)(38,46)(40,48)(41,43)(45,47)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=g,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, 9820114);