#8932764

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4063 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)(3,10)(4,6)(5,8)(7,9)(11,16)(12,13)(14,15)(19,25)(20,28)(23,26)(24,29)(33,34,39,36)(35,41,37,40)(38,42,46,44)(43,48,45,47);(1,3,7,5)(2,8,9,10)(4,14,11,12)(6,13,16,15)(17,18)(19,26)(20,29)(21,22)(23,25)(24,28)(27,31)(30,32)(33,35)(34,40)(36,41)(37,39)(38,43)(42,47)(44,48)(45,46);(2,9)(6,16)(8,10)(13,15)(18,22)(23,26)(24,29)(31,32)(33,34,39,36)(35,40,37,41)(38,44,46,42)(43,48,45,47);(1,4,7,11)(2,6,9,16)(3,12,5,14)(8,15,10,13)(17,19)(18,23)(20,30)(21,25)(22,26)(24,32)(27,28)(29,31)(33,36,39,34)(35,40,37,41)(38,44,46,42)(43,47,45,48);(1,5,7,3)(2,10,9,8)(4,12,11,14)(6,15,16,13)(17,20)(18,24)(19,27)(21,28)(22,29)(23,31)(25,30)(26,32)(33,37,39,35)(34,41,36,40)(38,45,46,43)(42,48,44,47);(1,6)(2,11)(3,13)(4,9)(5,15)(7,16)(8,12)(10,14)(17,18)(19,23)(20,24)(21,22)(25,26)(27,31)(28,29)(30,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839960);