#10408577

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	#42223	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3986 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)(3,4)(5,8)(6,9)(7,10)(11,12)(13,14)(15,16)(17,18,22,20)(19,25,27,23)(21,28,29,24)(26,30,32,31)(33,34)(35,40)(36,37)(38,41)(39,42)(43,46)(44,45)(47,48);(1,3)(2,4)(5,11)(6,7)(8,12)(9,10)(13,16)(14,15)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(33,34,39,42)(35,40,37,36)(38,41,47,48)(43,46,45,44);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,46,48,44);(1,2)(3,4)(5,13)(6,9)(7,10)(8,14)(11,16)(12,15)(17,20,22,18)(19,25,27,23)(21,24,29,28)(26,30,32,31)(33,36)(34,35)(37,42)(38,46)(39,40)(41,45)(43,48)(44,47);(1,4)(2,7)(3,9)(5,12)(6,10)(8,15)(11,13)(14,16)(17,19)(18,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,37,39,35)(34,40,42,36)(38,45,47,43)(41,46,48,44);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=egi,e^d=eghi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10315773);