#10121248

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	#56413	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4309 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)(3,6)(4,7)(5,10)(8,13)(9,12)(11,15)(14,16)(17,18)(19,23)(20,21)(22,24)(25,29)(26,27)(28,30)(31,32)(33,34,38,37)(35,41,43,39)(36,44,45,40)(42,46,48,47);(1,2,6,3)(4,9,12,7)(5,8,13,10)(11,15,16,14)(17,19,22,28)(18,23,24,30)(20,26,25,31)(21,27,29,32)(33,35)(34,39)(36,42)(37,41)(38,43)(40,46)(44,47)(45,48);(2,3)(7,9)(8,10)(14,15)(18,24)(20,25)(23,30)(26,31);(1,3,6,2)(4,9,12,7)(5,10,13,8)(11,15,16,14)(17,20,22,25)(18,21,24,29)(19,26,28,31)(23,27,30,32)(33,36,38,45)(34,40,37,44)(35,42,43,48)(39,46,41,47);(1,4,6,12)(2,7,3,9)(5,11,13,16)(8,14,10,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,30)(25,31)(29,32)(33,37,38,34)(35,41,43,39)(36,40,45,44)(42,46,48,47);(1,5)(2,8)(3,10)(4,11)(6,13)(7,14)(9,15)(12,16)(17,21)(18,25)(19,27)(20,24)(22,29)(23,31)(26,30)(28,32)(33,35)(34,41)(36,42)(37,39)(38,43)(40,47)(44,46)(45,48)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,d^2=ghi,e^2=g,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, 10028444);