#8416866

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	#57269	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4083 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8324062);