#8841404

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3888 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8748600);