#8933175

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	#51567	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4060 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)(3,7)(4,6)(5,8)(9,11)(10,14)(12,13)(15,16)(17,18,23,20)(19,25,21,24)(22,28,30,26)(27,31,29,32)(33,34,38,40)(35,39,43,48)(36,44,46,41)(37,47,45,42);(2,4)(3,9)(8,12)(10,15)(17,19,23,21)(18,24,20,25)(22,27,30,29)(26,31,28,32)(33,35,38,43)(34,39,40,48)(36,45,46,37)(41,47,44,42);(3,9)(5,13)(7,11)(8,12)(18,20)(22,30)(24,25)(27,29)(34,40)(35,43)(36,46)(42,47);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,16)(13,15)(17,20,23,18)(19,24,21,25)(22,28,30,26)(27,31,29,32)(33,36,38,46)(34,41,40,44)(35,37,43,45)(39,42,48,47);(1,4,6,2)(3,7,9,11)(5,12,13,8)(10,14,15,16)(17,21,23,19)(18,25,20,24)(22,29,30,27)(26,32,28,31)(33,37)(34,42)(35,36)(38,45)(39,41)(40,47)(43,46)(44,48);(1,5,6,13)(2,8,4,12)(3,10,9,15)(7,14,11,16)(17,22,23,30)(18,26,20,28)(19,27,21,29)(24,31,25,32)(33,34)(35,39)(36,41)(37,42)(38,40)(43,48)(44,46)(45,47)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,e^2=g,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,d^c=dgi,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, 8840371);