#8202380

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	#52369	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4104 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

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, 8109576);