#8527991

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	#2543	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3816 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)(3,6)(4,9)(5,8)(7,12)(10,13)(11,15)(14,16)(17,18,22,25)(19,23,28,31)(20,21,24,29)(26,27,30,32)(33,34,39,42)(35,37,41,38)(36,44,47,40)(43,46,48,45);(2,3)(7,9)(8,10)(14,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,35,39,41)(34,38,42,37)(36,43,47,48)(40,46,44,45);(1,2,6,3)(4,7,12,9)(5,8,13,10)(11,14,16,15)(17,19,22,28)(18,23,25,31)(20,26,24,30)(21,27,29,32)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20)(18,21)(19,26)(22,24)(23,27)(25,29)(28,30)(31,32)(33,36,39,47)(34,40,42,44)(35,43,41,48)(37,45,38,46);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37,39,38)(34,41,42,35)(36,45,47,46)(40,48,44,43);(1,5,6,13)(2,8,3,10)(4,11,12,16)(7,14,9,15)(17,21)(18,24)(19,27)(20,25)(22,29)(23,30)(26,31)(28,32)(33,38,39,37)(34,35,42,41)(36,46,47,45)(40,43,44,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=ghi,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=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, 8435187);