#8359689

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	#55806	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4115 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)(17,18,23,25)(19,26,20,21)(22,24,30,32)(27,31,28,29)(33,34)(35,40)(36,37)(38,41)(39,42)(43,47)(44,45)(46,48);(1,3,7,10)(2,8,4,6)(5,13,15,11)(9,14,12,16)(19,20)(21,26)(27,28)(29,31)(33,35,39,36)(34,37,42,40)(38,43,46,44)(41,45,48,47);(3,10)(6,8)(11,13)(14,16)(17,19,23,20)(18,21,25,26)(22,28,30,27)(24,31,32,29)(33,35,39,36)(34,40,42,37)(38,44,46,43)(41,45,48,47);(1,4,7,2)(3,6,10,8)(5,9,15,12)(11,16,13,14)(17,20,23,19)(18,21,25,26)(22,28,30,27)(24,29,32,31)(33,36,39,35)(34,40,42,37)(38,44,46,43)(41,47,48,45);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,16)(10,13)(17,21)(18,20)(19,25)(22,29)(23,26)(24,28)(27,32)(30,31)(33,37,39,40)(34,36,42,35)(38,45,46,47)(41,44,48,43);(1,6)(2,3)(4,10)(5,14)(7,8)(9,11)(12,13)(15,16)(17,22)(18,24)(19,27)(20,28)(21,29)(23,30)(25,32)(26,31)(33,38,39,46)(34,41,42,48)(35,43,36,44)(37,45,40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,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, 8266885);