#8927509

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

4220 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)(19,21)(20,25)(22,26)(23,27)(24,30)(28,29)(31,32)(33,34,39,37)(35,42,40,36)(38,45,47,41)(43,44,48,46);(1,3)(2,6)(4,5)(7,11)(8,14)(9,13)(10,16)(12,15)(17,19,23,30)(18,24,27,21)(20,28,26,32)(22,29,25,31)(33,35)(34,36)(37,42)(38,43)(39,40)(41,44)(45,46)(47,48);(3,11)(4,12)(6,16)(8,13)(20,26)(22,25)(28,32)(29,31)(33,36,39,42)(34,40,37,35)(38,46,47,44)(41,43,45,48);(1,4)(2,8)(3,5)(6,14)(7,12)(9,16)(10,13)(11,15)(17,20,23,26)(18,25,27,22)(19,28,30,32)(21,29,24,31)(33,36)(34,35)(37,40)(38,44)(39,42)(41,43)(45,48)(46,47);(1,5)(2,9)(3,12)(4,11)(6,8)(7,15)(10,14)(13,16)(17,21,23,24)(18,19,27,30)(20,31,26,29)(22,28,25,32)(33,37,39,34)(35,42,40,36)(38,45,47,41)(43,46,48,44);(1,6)(2,3)(4,14)(5,8)(7,16)(9,12)(10,11)(13,15)(17,22,23,25)(18,26,27,20)(19,29,30,31)(21,28,24,32)(33,38,39,47)(34,41,37,45)(35,43,40,48)(36,44,42,46)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=gh,e^2=g,f^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, 8834705);