#10202101

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3982 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)(3,11,14,8)(4,15,13,9)(6,10,12,16)(17,18,22,25)(19,24,28,23)(20,21,31,29)(26,32,30,27)(33,34,38,42)(35,43,45,39)(36,44,46,40)(37,41,48,47);(1,3)(2,8)(4,12)(5,11)(6,13)(7,14)(9,16)(10,15)(17,18,22,25)(19,24,28,23)(20,29,31,21)(26,27,30,32)(33,35,38,45)(34,39,42,43)(36,41,46,47)(37,40,48,44);(17,19,22,28)(18,23,25,24)(20,26,31,30)(21,27,29,32)(33,36,38,46)(34,40,42,44)(35,41,45,47)(37,39,48,43);(1,4,7,13)(2,9,5,15)(3,12,14,6)(8,16,11,10)(17,19)(18,24)(20,30)(21,27)(22,28)(23,25)(26,31)(29,32)(33,36)(34,40)(35,41)(37,43)(38,46)(39,48)(42,44)(45,47);(1,5,7,2)(3,11,14,8)(4,9,13,15)(6,10,12,16)(17,20,22,31)(18,21,25,29)(19,26,28,30)(23,27,24,32)(33,37)(34,41)(35,44)(36,39)(38,48)(40,45)(42,47)(43,46);(1,6,7,12)(2,10,5,16)(3,13,14,4)(8,15,11,9)(17,21)(18,20)(19,27)(22,29)(23,26)(24,30)(25,31)(28,32)(33,34)(35,39)(36,40)(37,47)(38,42)(41,48)(43,45)(44,46)

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=h,d^2=ghi,e^2=gh,f^2=ghi,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, 10109297);