#9910710

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	#49811	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3875 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,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, 9817906);