#8757317

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3773 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)(3,9)(4,7)(5,8)(6,12)(10,14)(11,15)(13,16)(18,23)(21,29)(24,30)(28,32)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,3,7,5)(2,8,4,9)(6,13,14,11)(10,16,12,15)(20,27)(24,30)(26,31)(28,32)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(2,4)(8,9)(10,12)(15,16)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,4,7,2)(3,9,5,8)(6,12,14,10)(11,16,13,15)(17,19,22,25)(18,21,23,29)(20,26,27,31)(24,28,30,32)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,5,7,3)(2,9,4,8)(6,13,14,11)(10,16,12,15)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,6,7,14)(2,10,4,12)(3,11,5,13)(8,15,9,16)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=ghi,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664513);