#8968105

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

4006 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=hi,e^2=h,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^c=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, 8875301);