#8925606

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4084 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:

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

Polycyclic presentation:

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

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, 8832802);