#8393770

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	#45917	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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)(17,18,22,21)(19,25,27,23)(20,28,30,24)(26,31,29,32)(33,34,39,41)(35,40,43,37)(36,42,45,47)(38,44,48,46);(2,3)(4,11)(8,10)(12,16)(17,19,22,27)(18,23,21,25)(20,26,30,29)(24,31,28,32)(33,35)(34,40)(36,38)(37,41)(39,43)(42,44)(45,48)(46,47);(5,13)(8,10)(12,16)(14,15)(18,21)(23,25)(24,28)(31,32)(34,41)(35,43)(38,48)(42,47);(1,3,6,2)(4,7,11,9)(5,10,13,8)(12,14,16,15)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,32)(27,29)(33,36)(34,42)(35,38)(37,46)(39,45)(40,44)(41,47)(43,48);(1,4,6,11)(2,7,3,9)(5,12,13,16)(8,14,10,15)(17,19)(18,23)(20,29)(21,25)(22,27)(24,32)(26,30)(28,31)(33,37,39,40)(34,43,41,35)(36,44,45,46)(38,47,48,42);(1,5,6,13)(2,8,3,10)(4,12,11,16)(7,14,9,15)(17,21,22,18)(19,25,27,23)(20,28,30,24)(26,32,29,31)(33,38)(34,44)(35,36)(37,47)(39,48)(40,42)(41,46)(43,45)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8300966);