#10201953

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4126 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,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, 10109149);