#8184804

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	#45868	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4067 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18,23,22)(19,26,20,24)(21,29,30,25)(27,31,28,32)(34,36)(39,42)(40,44)(46,48);(2,3)(7,9)(8,10)(14,15)(17,18,23,22)(19,26,20,24)(21,25,30,29)(27,32,28,31)(35,41)(39,42)(43,47)(46,48);(5,13)(8,10)(11,16)(14,15)(17,19,23,20)(18,24,22,26)(21,27,30,28)(25,31,29,32)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,23,19)(18,24,22,26)(21,28,30,27)(25,31,29,32)(33,35,38,41)(34,39,36,42)(37,43,45,47)(40,46,44,48);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,21,23,30)(18,25,22,29)(19,27,20,28)(24,31,26,32)(33,36,38,34)(35,42,41,39)(37,44,45,40)(43,48,47,46);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,22,23,18)(19,24,20,26)(21,29,30,25)(27,31,28,32)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8092000);