#9910900

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	#53185	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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=gh,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, 9818096);