#9720400

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	#45623	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4045 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)(3,5,9,13)(4,8,11,14)(10,12,15,16)(17,18)(19,27)(20,25)(21,31)(22,24)(23,28)(26,29)(30,32)(33,34,38,40)(35,36,42,44)(37,39,47,46)(41,43,45,48);(4,11)(8,14)(10,15)(12,16)(17,19,22,28)(18,23,24,27)(20,29,30,21)(25,31,32,26)(33,35)(34,36)(37,45)(38,42)(39,48)(40,44)(41,47)(43,46);(2,7)(3,9)(8,14)(10,15)(18,24)(23,27)(25,32)(26,31)(33,36,38,44)(34,35,40,42)(37,43,47,48)(39,41,46,45);(1,3)(2,5)(4,10)(6,9)(7,13)(8,12)(11,15)(14,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,31)(30,32)(33,36)(34,35)(37,43)(38,44)(39,41)(40,42)(45,46)(47,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,20,22,30)(18,25,24,32)(19,21,28,29)(23,26,27,31)(33,37)(34,39)(35,41)(36,43)(38,47)(40,46)(42,45)(44,48);(1,5)(2,3)(4,12)(6,13)(7,9)(8,10)(11,16)(14,15)(17,21)(18,26)(19,20)(22,29)(23,25)(24,31)(27,32)(28,30)(33,34)(35,36)(37,46)(38,40)(39,47)(41,48)(42,44)(43,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=i,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,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, 9627596);