#8927184

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	#2812	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3938 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)(3,8)(4,7)(5,9)(6,12)(10,14)(11,13)(15,16)(17,18)(19,22)(20,29)(21,25)(23,26)(24,28)(27,32)(30,31)(33,34,39,44)(35,42,37,40)(36,46,47,41)(38,43,45,48);(1,3)(2,8)(4,5)(6,11)(7,9)(10,15)(12,13)(14,16)(17,19)(18,22)(20,21)(23,27)(24,30)(25,29)(26,32)(28,31)(33,35,39,37)(34,40,44,42)(36,38,47,45)(41,43,46,48);(2,4)(3,9)(6,14)(13,15)(19,27)(20,28)(22,32)(24,29)(33,36,39,47)(34,41,44,46)(35,45,37,38)(40,48,42,43);(1,4,7,2)(3,5,9,8)(6,12,14,10)(11,13,16,15)(17,20)(18,24)(19,21)(22,30)(23,28)(25,32)(26,29)(27,31)(33,37,39,35)(34,42,44,40)(36,38,47,45)(41,43,46,48);(1,5)(2,9)(3,4)(6,13)(7,8)(10,16)(11,12)(14,15)(17,21)(18,25)(19,28)(20,27)(22,24)(23,31)(26,30)(29,32)(33,38)(34,43)(35,36)(37,47)(39,45)(40,41)(42,46)(44,48);(1,6)(2,10)(3,11)(4,12)(5,13)(7,14)(8,15)(9,16)(17,22)(18,19)(20,30)(21,24)(23,32)(25,28)(26,27)(29,31)(33,34)(35,40)(36,41)(37,42)(38,43)(39,44)(45,48)(46,47)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=h,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, 8834380);