#10195770

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	#56088	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3901 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=h,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, 10102966);