#8899612

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4022 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=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, 8806808);