#8416978

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	#53533	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3983 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=i,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324174);