#8188007

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	#40951	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4127 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,14)(15,16)(17,18,22,26)(19,24,20,23)(21,25,29,32)(27,31,28,30)(33,34)(35,43)(36,47)(37,48)(38,42)(39,40)(41,44)(45,46);(2,7)(3,10)(9,14)(11,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(34,40)(35,44)(36,48)(42,46);(5,13)(9,14)(11,16)(12,15)(17,19,22,20)(18,23,26,24)(21,28,29,27)(25,31,32,30)(33,35,39,44)(34,41,40,43)(36,45,48,38)(37,46,47,42);(1,3)(2,8)(4,7)(5,11)(6,10)(9,15)(12,14)(13,16)(17,20,22,19)(18,24,26,23)(21,28,29,27)(25,31,32,30)(33,36)(34,37)(35,45)(38,44)(39,48)(40,47)(41,46)(42,43);(1,4,6,8)(2,3,7,10)(5,12,13,15)(9,11,14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32)(33,37)(34,36)(35,46)(38,43)(39,47)(40,48)(41,45)(42,44);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,21,22,29)(18,25,26,32)(19,27,20,28)(23,30,24,31)(33,38,39,45)(34,42,40,46)(35,36,44,48)(37,43,47,41)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=hi,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, 8095203);