#8535025

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	#2408	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4170 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=hi,d^2=g,e^2=ghi,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, 8442221);