#10457915

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	#48089	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4101 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:

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

Polycyclic presentation:

2^9|a^2=ghi,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, 10365111);