#9726631

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	#53905	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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)(3,4)(5,13)(6,8)(7,11)(9,15)(10,16)(12,14)(17,18)(19,24)(20,23)(21,25)(22,26)(27,30)(28,31)(29,32)(34,39)(35,42)(37,45)(44,47);(1,3,6,11)(2,7,8,4)(5,10,15,14)(9,16,13,12)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,32,28,30)(33,34,38,39)(35,36,42,40)(37,41,45,46)(43,44,48,47);(2,8)(4,7)(9,13)(12,16)(18,20)(19,26)(25,28)(27,29)(33,35,38,42)(34,40,39,36)(37,43,45,48)(41,47,46,44);(1,4)(2,3)(5,12)(6,7)(8,11)(9,10)(13,14)(15,16)(17,20,23,18)(19,22,26,24)(21,28,31,25)(27,32,29,30)(33,35)(34,36)(37,43)(38,42)(39,40)(41,44)(45,48)(46,47);(1,3)(2,4)(5,14)(6,11)(7,8)(9,16)(10,15)(12,13)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,32,24,30)(33,36)(34,35)(37,44)(38,40)(39,42)(41,43)(45,47)(46,48);(1,5)(2,9)(3,10)(4,12)(6,15)(7,16)(8,13)(11,14)(17,22)(18,19)(20,26)(21,30)(23,24)(25,29)(27,28)(31,32)(33,37,38,45)(34,41,39,46)(35,43,42,48)(36,44,40,47)

Polycyclic presentation:

2^9|b^2=h,c^2=ghi,d^2=gh,e^2=gh,f^2=ghi,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, 9633827);