#8925010

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

4243 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^2=g,f^2=g,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, 8832206);