#8745928

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	#43444	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3943 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)(3,7)(4,10)(5,9)(6,8)(11,14)(12,13)(15,16)(17,18)(19,28)(20,25)(21,32)(22,27)(23,30)(24,31)(26,29)(33,34,38,42)(35,37,41,39)(36,40,44,47)(43,45,48,46);(1,3,6,10)(2,7,8,4)(5,11,15,13)(9,14,16,12)(17,19)(18,23)(20,21)(22,30)(24,32)(25,26)(27,28)(29,31)(36,44)(40,47)(43,48)(45,46);(2,8)(3,10)(9,16)(11,13)(17,20,22,31)(18,24,27,25)(19,21,30,29)(23,32,28,26)(33,35,38,41)(34,39,42,37)(36,43,44,48)(40,46,47,45);(1,3)(2,4)(5,13)(6,10)(7,8)(9,14)(11,15)(12,16)(17,20)(18,25)(19,29)(21,30)(22,31)(23,32)(24,27)(26,28)(33,36,38,44)(34,40,42,47)(35,43,41,48)(37,45,39,46);(1,4)(2,3)(5,14)(6,7)(8,10)(9,13)(11,16)(12,15)(17,21,22,29)(18,26,27,32)(19,20,30,31)(23,25,28,24)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,5,6,15)(2,9,8,16)(3,11,10,13)(4,12,7,14)(17,18)(19,23)(20,24)(21,32)(22,27)(25,31)(26,29)(28,30)(33,37,38,39)(34,41,42,35)(36,45,44,46)(40,48,47,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=hi,d^2=ghi,e^2=g,f^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, 8653124);