#8393757

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

4115 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)(3,5)(4,12)(6,8)(7,11)(9,15)(10,16)(13,14)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30)(33,34,39,41)(35,40,43,37)(36,42,45,47)(38,44,48,46);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,35)(34,40)(36,38)(37,41)(39,43)(42,44)(45,48)(46,47);(2,8)(5,7)(9,12)(14,16)(21,29)(24,31)(26,32)(28,30)(34,41)(35,43)(38,48)(42,47);(1,3)(2,7)(4,13)(5,8)(6,11)(9,14)(10,15)(12,16)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,36)(34,42)(35,38)(37,46)(39,45)(40,44)(41,47)(43,48);(1,4,6,15)(2,9,8,12)(3,10,11,13)(5,14,7,16)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31)(33,37,39,40)(34,43,41,35)(36,44,45,46)(38,47,48,42);(1,5,6,7)(2,3,8,11)(4,14,15,16)(9,10,12,13)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,38)(34,44)(35,36)(37,47)(39,48)(40,42)(41,46)(43,45)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,d^2=ghi,e^2=h,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, 8300953);