#8928712

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	#46505	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4042 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,e^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, 8835908);