#9815134

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	#43375	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=ghi,e^2=g,f^2=gh,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, 9722330);