#8754827

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	#2673	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3850 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8662023);