#8927378

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	#53508	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4128 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=h,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, 8834574);