#9910079

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	#53177	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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)(3,6)(4,11)(5,8)(7,12)(9,13)(10,16)(14,15)(17,18)(19,24)(20,28)(21,22)(23,27)(25,31)(26,32)(29,30)(33,34)(35,37)(36,40)(38,41)(39,42)(43,45)(44,46)(47,48);(3,9)(6,13)(10,15)(14,16)(17,19)(18,24)(20,21)(22,28)(23,29)(25,32)(26,31)(27,30)(33,35,39,36)(34,40,42,37)(38,43,47,44)(41,46,48,45);(1,2,5,8)(3,6,9,13)(4,7,12,11)(10,14,15,16)(17,20,23,31)(18,25,27,28)(19,26,29,21)(22,30,32,24)(33,36,39,35)(34,40,42,37)(38,44,47,43)(41,46,48,45);(1,2)(3,6)(4,7)(5,8)(9,13)(10,14)(11,12)(15,16)(17,20)(18,25)(19,21)(22,30)(23,31)(24,32)(26,29)(27,28);(1,3)(2,6)(4,10)(5,9)(7,14)(8,13)(11,16)(12,15)(17,21)(18,22)(19,20)(23,26)(24,28)(25,30)(27,32)(29,31)(33,37,39,40)(34,35,42,36)(38,46,47,45)(41,44,48,43);(1,4)(2,7)(3,10)(5,12)(6,14)(8,11)(9,15)(13,16)(17,22)(18,26)(19,28)(20,30)(21,27)(23,32)(24,31)(25,29)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,46)(42,48)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9817275);