#9863962

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	#55266	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3959 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)(3,10)(4,7)(5,9)(6,8)(11,16)(12,13)(14,15)(17,18,22,25)(19,23,27,20)(21,29,31,24)(26,28,32,30);(1,3,7,6)(2,8,4,10)(5,11,13,14)(9,15,12,16)(17,18,22,25)(19,23,27,20)(21,24,31,29)(26,30,32,28)(33,34,39,36)(35,42,40,37)(38,41,47,44)(43,46,48,45);(2,4)(8,10)(9,12)(15,16)(17,19,22,27)(18,20,25,23)(21,26,31,32)(24,28,29,30)(33,35)(34,37)(36,42)(38,43)(39,40)(41,45)(44,46)(47,48);(1,4,7,2)(3,10,6,8)(5,12,13,9)(11,16,14,15)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,29)(31,32)(33,36,39,34)(35,42,40,37)(38,44,47,41)(43,46,48,45);(1,5)(2,9)(3,11)(4,12)(6,14)(7,13)(8,15)(10,16)(17,20,22,23)(18,19,25,27)(21,30,31,28)(24,32,29,26)(33,37,39,42)(34,40,36,35)(38,46,47,45)(41,43,44,48);(1,6,7,3)(2,10,4,8)(5,11,13,14)(9,15,12,16)(17,21)(18,24)(19,26)(20,28)(22,31)(23,30)(25,29)(27,32)(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=gi,c^2=gh,d^2=hi,e^2=i,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, 9771158);