#8359735

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	#41165	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4095 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=gh,d^2=gi,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,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, 8266931);