#8772475

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	#49205	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3996 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)(4,12)(5,6)(8,13)(9,10)(11,15)(14,16)(17,18,23,27)(19,26,21,24)(20,22,31,25)(28,30,29,32)(34,36)(35,43)(38,48)(41,45);(1,2)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,19,23,21)(18,24,27,26)(20,28,31,29)(22,32,25,30)(33,34,39,36)(35,44,43,40)(37,45,47,41)(38,42,48,46);(2,6)(3,10)(8,12)(11,16)(17,20,23,31)(18,25,27,22)(19,28,21,29)(24,30,26,32)(33,35)(34,40)(36,44)(37,38)(39,43)(41,42)(45,46)(47,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,21,23,19)(18,26,27,24)(20,28,31,29)(22,32,25,30)(33,36,39,34)(35,44,43,40)(37,45,47,41)(38,46,48,42);(1,3,5,10)(2,7,6,9)(4,11,13,16)(8,14,12,15)(17,20)(18,22)(19,29)(21,28)(23,31)(24,30)(25,27)(26,32)(33,37)(34,41)(35,38)(36,45)(39,47)(40,42)(43,48)(44,46);(1,4,5,13)(2,8,6,12)(3,11,10,16)(7,14,9,15)(17,22)(18,20)(19,30)(21,32)(23,25)(24,29)(26,28)(27,31)(33,38)(34,42)(35,37)(36,46)(39,48)(40,41)(43,47)(44,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=hi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8679671);