#9863967

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	#54018	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3952 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)(5,9)(6,7)(8,11)(10,12)(13,16)(14,15)(17,18,22,25)(19,23,27,20)(21,29,31,24)(26,28,32,30)(33,34,40,44)(35,45,48,41)(36,37,38,42)(39,47,46,43);(2,7)(4,8)(9,15)(12,13)(17,18,22,25)(19,23,27,20)(21,24,31,29)(26,30,32,28)(33,35)(34,41)(36,43)(37,46)(38,47)(39,42)(40,48)(44,45);(17,19,22,27)(18,20,25,23)(21,26,31,32)(24,28,29,30)(33,36,40,38)(34,42,44,37)(35,43,48,47)(39,45,46,41);(1,3)(2,8)(4,7)(5,10)(6,11)(9,13)(12,15)(14,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,29)(31,32)(33,37,40,42)(34,36,44,38)(35,46,48,39)(41,43,45,47);(1,4,6,8)(2,3,7,11)(5,13,14,12)(9,16,15,10)(17,20,22,23)(18,19,25,27)(21,30,31,28)(24,32,29,26)(33,38,40,36)(34,42,44,37)(35,47,48,43)(39,45,46,41);(1,5,6,14)(2,9,7,15)(3,10,11,16)(4,12,8,13)(17,21)(18,24)(19,26)(20,28)(22,31)(23,30)(25,29)(27,32)(33,39,40,46)(34,43,44,47)(35,42,48,37)(36,45,38,41)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,d^2=hi,e^2=gi,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, 9771163);