#9910707

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	#53238	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3879 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)(3,8)(4,9)(5,10)(6,7)(11,13)(12,16)(14,15)(17,18,23,21)(19,27,29,24)(20,30,31,25)(22,32,28,26)(35,41)(39,45)(42,47)(46,48);(2,7)(4,8)(10,15)(12,13)(17,19,23,29)(18,24,21,27)(20,28,31,22)(25,26,30,32)(34,38)(39,45)(40,43)(46,48);(17,20,23,31)(18,25,21,30)(19,22,29,28)(24,32,27,26)(33,34)(35,39)(36,40)(37,38)(41,45)(42,46)(43,44)(47,48);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,13,15,12)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,26)(27,32)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,4,6,8)(2,9,7,3)(5,13,14,12)(10,11,15,16)(17,21,23,18)(19,24,29,27)(20,30,31,25)(22,32,28,26)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,5,6,14)(2,10,7,15)(3,11,9,16)(4,12,8,13)(17,22,23,28)(18,26,21,32)(19,20,29,31)(24,30,27,25)(33,36)(34,40)(35,42)(37,44)(38,43)(39,46)(41,47)(45,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=h,d^2=gh,e^2=g,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, 9817903);