#8097079

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	#2414	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3945 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)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,18,23,24)(19,26,20,25)(21,27,31,30)(22,28,32,29)(33,34,38,36)(35,39,41,42)(37,40,45,44)(43,46,47,48);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(18,24)(19,20)(27,30)(28,29)(35,41)(39,42)(43,47)(46,48);(17,19,23,20)(18,25,24,26)(21,29,31,28)(22,27,32,30)(33,35)(34,39)(36,42)(37,43)(38,41)(40,46)(44,48)(45,47);(1,4,5,13)(2,7,9,12)(3,10,11,16)(6,14,8,15)(17,20,23,19)(18,26,24,25)(21,28,31,29)(22,30,32,27);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,21)(18,27)(19,29)(20,28)(22,26)(23,31)(24,30)(25,32)(33,36,38,34)(35,42,41,39)(37,40,45,44)(43,46,47,48);(1,2)(3,6)(4,7)(5,9)(8,11)(10,14)(12,13)(15,16)(17,22)(18,28)(19,27)(20,30)(21,25)(23,32)(24,29)(26,31)(33,37,38,45)(34,40,36,44)(35,43,41,47)(39,46,42,48)

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004275);