#9782765

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	#49875	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4035 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,d^2=g,e^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, 9689961);