#10106453

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	#40148	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4005 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=hi,d^2=g,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, 10013649);