#8158015

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	#36886	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4112 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=hi,d^2=g,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi,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, 8065211);