#10457914

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4160 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)(4,8)(5,7)(6,9)(11,15)(12,14)(13,16)(17,18,23,28)(19,24,30,32)(20,21,26,25)(22,31,29,27)(33,34)(35,44)(36,42)(37,48)(38,46)(39,43)(40,47)(41,45);(1,3,6,5)(2,7,9,10)(4,12,13,11)(8,15,16,14)(17,19,23,30)(18,24,28,32)(20,29,26,22)(21,27,25,31)(33,35)(34,38)(36,37)(39,46)(40,48)(41,42)(43,44)(45,47);(17,20)(18,25)(19,29)(21,28)(22,30)(23,26)(24,31)(27,32)(33,36)(34,40)(35,45)(37,46)(38,41)(39,47)(42,43)(44,48);(1,4,6,13)(2,8,9,16)(3,11,5,12)(7,14,10,15)(17,20)(18,21)(19,29)(22,30)(23,26)(24,27)(25,28)(31,32)(33,37,39,45)(34,41,43,48)(35,36,46,47)(38,42,44,40);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,21)(18,26)(19,27)(20,28)(22,24)(23,25)(29,32)(30,31)(33,36)(34,42)(35,37)(38,41)(39,47)(40,43)(44,48)(45,46);(1,5,6,3)(2,7,9,10)(4,12,13,11)(8,14,16,15)(17,22,23,29)(18,27,28,31)(19,20,30,26)(21,24,25,32)(33,38)(34,35)(36,41)(37,42)(39,44)(40,45)(43,46)(47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10365110);