#9863972

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	#2714	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3797 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=hi,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,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, 9771168);