#8649882

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3894 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)(3,10)(4,12)(5,6)(7,8)(9,15)(11,14)(13,16)(17,18,22,23)(19,24,30,27)(20,21,26,25)(28,31,32,29);(1,3,7,6)(2,5,8,10)(4,13,15,11)(9,16,12,14)(17,18,22,23)(19,27,30,24)(20,21,26,25)(28,29,32,31)(33,34,38,41)(35,42,36,39)(37,40,45,48)(43,47,44,46);(2,8)(5,10)(9,12)(14,16)(18,23)(20,26)(24,27)(28,32)(33,35,38,36)(34,39,41,42)(37,43,45,44)(40,46,48,47);(1,4,7,15)(2,9,8,12)(3,11,6,13)(5,14,10,16)(17,19)(18,24)(20,28)(21,31)(22,30)(23,27)(25,29)(26,32)(33,36,38,35)(34,39,41,42)(37,43,45,44)(40,47,48,46);(1,5)(2,3)(4,14)(6,8)(7,10)(9,11)(12,13)(15,16)(17,20,22,26)(18,25,23,21)(19,28,30,32)(24,29,27,31)(33,34)(35,42)(36,39)(37,40)(38,41)(43,47)(44,46)(45,48);(1,6,7,3)(2,5,8,10)(4,11,15,13)(9,16,12,14)(17,21,22,25)(18,26,23,20)(19,29,30,31)(24,28,27,32)(33,37,38,45)(34,40,41,48)(35,43,36,44)(39,46,42,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8557078);