#8893574

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	#46777	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4078 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=gh,e^2=g,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, 8800770);