#8728062

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3869 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)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18)(19,27)(20,31)(21,25)(22,26)(23,30)(24,29)(28,32)(33,34,39,40)(35,36,43,48)(37,41,38,42)(44,46,45,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,32)(25,31)(26,27)(34,40)(35,43)(41,42)(44,45);(3,10)(6,7)(11,16)(14,15)(17,18,22,26)(19,23,30,27)(20,24,29,31)(21,25,28,32)(33,35)(34,36)(37,44)(38,45)(39,43)(40,48)(41,46)(42,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,18)(19,27)(20,24)(21,32)(22,26)(23,30)(25,28)(29,31)(33,36)(34,35)(37,46)(38,47)(39,48)(40,43)(41,44)(42,45);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,20)(18,24)(19,28)(21,30)(22,29)(23,32)(25,27)(26,31)(33,37,39,38)(34,41,40,42)(35,44,43,45)(36,46,48,47);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21,22,28)(18,25,26,32)(19,29,30,20)(23,31,27,24)(33,38,39,37)(34,42,40,41)(35,45,43,44)(36,47,48,46)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8635258);