#10202087

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4089 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18,21,24)(19,25,27,22)(20,23,29,32)(26,31,28,30)(33,34)(35,46)(36,39)(37,47)(38,43)(40,45)(41,44)(42,48);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19)(18,22)(20,28)(21,27)(23,31)(24,25)(26,29)(30,32)(33,35)(34,38)(36,47)(37,41)(39,48)(40,43)(42,44)(45,46);(17,19,21,27)(18,22,24,25)(20,26,29,28)(23,30,32,31)(33,36,40,44)(34,41,45,39)(35,47,43,42)(37,46,48,38);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,29)(31,32)(33,37,40,48)(34,42,45,47)(35,41,43,39)(36,38,44,46);(1,5,6,3)(2,7,9,10)(4,11,13,14)(8,16,12,15)(17,20)(18,23)(19,26)(21,29)(22,30)(24,32)(25,31)(27,28)(33,38)(34,43)(35,45)(36,37)(39,47)(40,46)(41,42)(44,48);(1,2)(3,7)(4,8)(5,10)(6,9)(11,15)(12,13)(14,16)(17,18)(19,22)(20,23)(21,24)(25,27)(26,30)(28,31)(29,32)(33,39,40,41)(34,44,45,36)(35,48,43,37)(38,42,46,47)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=hi,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

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, 10109283);