#9910030

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	#48050	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,f^2=g,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, 9817226);