#8925965

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

4147 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)(17,18,22,23)(19,21,27,24)(20,25,30,31)(26,29,28,32)(33,34,40,38)(35,43,42,39)(36,46,37,41)(44,48,45,47);(1,2)(3,7)(4,11)(5,6)(8,13)(9,14)(10,16)(12,15)(17,18,22,23)(19,24,27,21)(20,25,30,31)(26,32,28,29)(33,35)(34,39)(36,44)(37,45)(38,43)(40,42)(41,48)(46,47);(2,6)(7,14)(8,11)(15,16)(18,23)(21,24)(25,31)(29,32)(33,36,40,37)(34,41,38,46)(35,44,42,45)(39,48,43,47);(1,3)(2,7)(4,12)(5,9)(6,14)(8,16)(10,13)(11,15)(17,19)(18,24)(20,28)(21,23)(22,27)(25,29)(26,30)(31,32)(33,37,40,36)(34,41,38,46)(35,45,42,44)(39,48,43,47);(1,4)(2,8)(3,10)(5,13)(6,11)(7,15)(9,12)(14,16)(17,20)(18,25)(19,26)(21,29)(22,30)(23,31)(24,32)(27,28)(33,38,40,34)(35,43,42,39)(36,46,37,41)(44,47,45,48);(1,2)(3,7)(4,8)(5,6)(9,14)(10,15)(11,13)(12,16)(17,21,22,24)(18,19,23,27)(20,29,30,32)(25,26,31,28)(33,39,40,43)(34,42,38,35)(36,47,37,48)(41,44,46,45)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=g,e^2=g,f^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, 8833161);