#8901600

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	#40139	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3976 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=g,d^2=hi,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8808796);