#10409768

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	#2688	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3985 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)(5,8)(6,9)(7,10)(11,12)(13,14)(15,16)(17,18)(19,25)(20,30)(21,26)(22,23)(24,28)(27,29)(31,32)(33,34,38,41)(35,36,43,39)(37,40,46,48)(42,44,47,45);(1,3)(2,4)(5,11)(6,7)(8,12)(9,10)(13,16)(14,15)(17,19,24,31)(18,25,28,32)(20,29,21,23)(22,30,27,26)(33,35,38,43)(34,39,41,36)(37,42,46,47)(40,45,48,44);(1,2)(3,10)(4,7)(5,8)(6,9)(11,16)(12,15)(13,14)(17,20,24,21)(18,26,28,30)(19,29,31,23)(22,32,27,25)(33,34,38,41)(35,39,43,36)(37,40,46,48)(42,45,47,44);(1,2)(3,4)(5,13)(6,9)(7,10)(8,14)(11,16)(12,15)(17,21,24,20)(18,26,28,30)(19,23,31,29)(22,32,27,25)(33,36)(34,35)(37,45)(38,39)(40,47)(41,43)(42,48)(44,46);(1,4)(2,7)(3,9)(5,12)(6,10)(8,15)(11,13)(14,16)(17,22,24,27)(18,23,28,29)(19,30,31,26)(20,32,21,25)(33,35)(34,39)(36,41)(37,42)(38,43)(40,45)(44,48)(46,47);(1,5)(2,8)(3,11)(4,12)(6,14)(7,15)(9,13)(10,16)(17,23)(18,27)(19,20)(21,31)(22,28)(24,29)(25,26)(30,32)(33,37)(34,40)(35,42)(36,44)(38,46)(39,45)(41,48)(43,47)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=h,e^2=h,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, 10316964);