#8734521

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	#38509	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4100 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)(3,7)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18,22,26)(19,25,30,23)(20,21,31,24)(27,32,28,29)(34,39)(35,41)(36,44)(45,46);(1,3,6,11)(2,7,10,16)(4,5,12,15)(8,14,13,9)(17,19)(18,23)(20,27)(21,29)(22,30)(24,32)(25,26)(28,31)(34,39)(35,41)(40,47)(43,48);(3,11)(4,12)(7,16)(8,13)(17,20,22,31)(18,24,26,21)(19,27,30,28)(23,32,25,29)(33,34,38,39)(35,42,41,37)(36,40,44,47)(43,45,48,46);(1,4)(2,8)(3,5)(6,12)(7,14)(9,16)(10,13)(11,15)(17,20)(18,21)(19,28)(22,31)(23,32)(24,26)(25,29)(27,30)(33,35,38,41)(34,37,39,42)(36,43,44,48)(40,46,47,45);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,16)(12,13)(17,21)(18,20)(19,29)(22,24)(23,27)(25,28)(26,31)(30,32)(33,36)(34,40)(35,43)(37,46)(38,44)(39,47)(41,48)(42,45);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,19)(18,25)(20,27)(21,32)(22,30)(23,26)(24,29)(28,31)(33,37)(34,35)(36,45)(38,42)(39,41)(40,48)(43,47)(44,46)

Polycyclic presentation:

2^9|a^2=g,b^2=ghi,c^2=h,d^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8641717);