#8927040

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	#46447	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4150 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)(3,6)(4,13)(5,9)(7,10)(8,12)(11,16)(14,15)(17,18,22,25)(19,23,28,24)(20,21,30,32)(26,27,29,31)(33,34)(35,36)(37,45)(38,39)(40,41)(42,46)(43,47)(44,48);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19)(18,23)(20,26)(21,27)(22,28)(24,25)(29,30)(31,32)(34,39)(35,41)(42,45)(43,48);(3,11)(4,12)(6,16)(8,13)(17,19,22,28)(18,24,25,23)(20,26,30,29)(21,31,32,27)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,19)(18,24)(20,29)(21,27)(22,28)(23,25)(26,30)(31,32)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,20)(18,21)(19,26)(22,30)(23,27)(24,31)(25,32)(28,29)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,21)(18,20)(19,27)(22,32)(23,26)(24,29)(25,30)(28,31)(33,37)(34,42)(35,43)(36,44)(38,46)(39,45)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,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, 8834236);