#8139653

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4092 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=gh,d^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,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, 8046849);