#8994405

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3786 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)(3,8)(4,10)(5,11)(6,14)(7,9)(12,16)(13,15)(33,34,40,44)(35,45,47,38)(36,46,48,41)(37,43,39,42);(1,3,7,4)(2,8,9,10)(5,14,15,12)(6,13,16,11)(17,18,22,26)(19,24,29,23)(20,21,25,32)(27,31,30,28)(33,35,40,47)(34,38,44,45)(36,43,48,42)(37,41,39,46);(2,9)(3,4)(5,15)(6,16)(17,19,22,29)(18,23,26,24)(20,30,25,27)(21,31,32,28)(33,36,40,48)(34,41,44,46)(35,42,47,43)(37,45,39,38);(1,4,7,3)(2,10,9,8)(5,12,15,14)(6,11,16,13)(17,20,22,25)(18,21,26,32)(19,27,29,30)(23,28,24,31)(33,37,40,39)(34,42,44,43)(35,41,47,46)(36,45,48,38);(1,5,7,15)(2,11,9,13)(3,12,4,14)(6,8,16,10)(17,19)(18,24)(20,30)(21,28)(22,29)(23,26)(25,27)(31,32)(33,38)(34,35)(36,37)(39,48)(40,45)(41,42)(43,46)(44,47);(1,6)(2,12)(3,13)(4,11)(5,8)(7,16)(9,14)(10,15)(17,21)(18,25)(19,28)(20,26)(22,32)(23,30)(24,27)(29,31)(33,39,40,37)(34,43,44,42)(35,46,47,41)(36,38,48,45)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8901601);