#9910721

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	#53174	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18)(19,25)(20,24)(21,26)(22,23)(27,28)(29,31)(30,32)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(3,9)(7,13)(10,15)(14,16)(17,18)(19,21)(20,24)(22,23)(25,26)(27,30)(28,32)(29,31)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(2,6)(7,13)(8,11)(14,16)(18,23)(19,26)(24,31)(27,32)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,19)(18,21)(20,27)(22,26)(23,25)(24,30)(28,31)(29,32);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,20,22,29)(18,24,23,31)(19,27,26,32)(21,30,25,28)(33,34)(35,42)(36,39)(37,45)(38,41)(40,46)(43,47)(44,48);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,21)(18,19)(20,28)(22,25)(23,26)(24,32)(27,31)(29,30)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|b^2=g,c^2=g,e^2=h,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^e=fgi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817917);