#8757404

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

3957 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,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, 8664600);