#8371248

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3849 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,d^2=ghi,e^2=hi,f^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,e^d=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, 8278444);