#8932750

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	#47700	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4066 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,f^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, 8839946);