#8773014

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	#53297	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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)(3,12)(4,13)(5,11)(6,16)(7,9)(8,14)(10,15)(17,18,23,20)(19,26,22,24)(21,25,30,28)(27,32,29,31)(33,34,39,43)(35,37,46,40)(36,47,45,41)(38,48,44,42);(1,3,7,14)(2,8,9,12)(4,5,15,16)(6,10,11,13)(17,18,23,20)(19,24,22,26)(21,28,30,25)(27,32,29,31)(33,35,39,46)(34,40,43,37)(36,38,45,44)(41,48,47,42);(2,9)(4,15)(5,16)(8,12)(17,19,23,22)(18,24,20,26)(21,27,30,29)(25,31,28,32)(33,36,39,45)(34,41,43,47)(35,44,46,38)(37,48,40,42);(1,4)(2,10)(3,5)(6,12)(7,15)(8,11)(9,13)(14,16)(17,20,23,18)(19,26,22,24)(21,28,30,25)(27,32,29,31)(33,37)(34,35)(36,48)(38,47)(39,40)(41,44)(42,45)(43,46);(1,5)(2,11)(3,4)(6,9)(7,16)(8,10)(12,13)(14,15)(17,21,23,30)(18,25,20,28)(19,27,22,29)(24,31,26,32)(33,35)(34,37)(36,44)(38,45)(39,46)(40,43)(41,48)(42,47);(1,6,7,11)(2,5,9,16)(3,13,14,10)(4,12,15,8)(17,22,23,19)(18,26,20,24)(21,29,30,27)(25,32,28,31)(33,38,39,44)(34,42,43,48)(35,45,46,36)(37,47,40,41)

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=ghi,d^2=gh,e^2=gh,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=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, 8680210);