#8756051

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	#2428	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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:

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

Polycyclic presentation:

2^9|a^2=gh,d^2=g,e^2=ghi,f^2=h,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, 8663247);