#10202012

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	#40980	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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)(3,10)(4,13)(5,12)(6,8)(7,14)(9,15)(11,16)(17,18,23,25)(19,24,27,31)(20,22,29,32)(21,26,28,30)(33,34)(35,40)(36,42)(37,43)(38,48)(39,41)(44,45)(46,47);(1,3,6,14)(2,7,8,10)(4,12,15,11)(5,13,16,9)(17,19,23,27)(18,24,25,31)(20,28,29,21)(22,30,32,26)(33,35,39,38)(34,40,41,48)(36,44,43,46)(37,47,42,45);(2,8)(4,15)(7,10)(11,12)(18,25)(19,27)(20,29)(26,30)(34,41)(36,43)(40,48)(44,46);(1,4)(2,9)(3,12)(5,7)(6,15)(8,13)(10,16)(11,14)(17,20)(18,22)(19,28)(21,27)(23,29)(24,30)(25,32)(26,31)(33,36,39,43)(34,42,41,37)(35,44,38,46)(40,45,48,47);(1,3)(2,10)(4,11)(5,9)(6,14)(7,8)(12,15)(13,16)(17,21,23,28)(18,26,25,30)(19,29,27,20)(22,24,32,31)(33,37,39,42)(34,43,41,36)(35,45,38,47)(40,44,48,46);(1,5)(2,11)(3,13)(4,7)(6,16)(8,12)(9,14)(10,15)(17,22)(18,20)(19,30)(21,31)(23,32)(24,28)(25,29)(26,27)(33,38,39,35)(34,40,41,48)(36,46,43,44)(37,47,42,45)

Polycyclic presentation:

2^9|a^2=i,b^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, 10109208);