#8928750

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	#45685	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3855 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=h,d^2=g,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=dgi,f^c=fgi,e^d=egi

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, 8835946);