#10420075

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	#54852	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4123 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)(6,10)(7,8)(11,12)(13,15)(14,16)(17,18,23,25)(19,26,20,24)(21,22,31,32)(27,30,29,28)(33,34,38,41)(35,39,44,37)(36,40,46,48)(42,47,43,45);(1,2,6,10)(3,7,8,4)(5,9,14,16)(11,15,13,12)(33,35,38,44)(34,39,41,37)(36,43,46,42)(40,45,48,47);(17,19,23,20)(18,24,25,26)(21,27,31,29)(22,28,32,30)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,3)(2,7)(4,10)(5,13)(6,8)(9,12)(11,14)(15,16)(17,20,23,19)(18,24,25,26)(21,27,31,29)(22,30,32,28)(33,36)(34,40)(35,43)(37,47)(38,46)(39,45)(41,48)(42,44);(1,4,6,7)(2,8,10,3)(5,12,14,15)(9,13,16,11)(17,21)(18,22)(19,27)(20,29)(23,31)(24,28)(25,32)(26,30)(33,35)(34,37)(36,42)(38,44)(39,41)(40,45)(43,46)(47,48);(1,5,6,14)(2,9,10,16)(3,11,8,13)(4,12,7,15)(17,22)(18,21)(19,28)(20,30)(23,32)(24,27)(25,31)(26,29)(33,37)(34,35)(36,45)(38,39)(40,42)(41,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gi,c^2=h,d^2=h,e^2=g,f^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, 10327271);