#8119256

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	#57172	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4049 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=ghi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^c=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, 8026452);