#9782709

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	#41038	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4170 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,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, 9689905);