#8334352

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3912 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)(3,11)(4,14)(5,8)(6,12)(7,9)(10,15)(13,16)(17,18)(19,23)(20,28)(21,22)(24,30)(25,27)(26,32)(29,31)(33,34,39,36)(35,38,41,42)(37,40,46,44)(43,45,47,48);(1,3,7,5)(2,8,9,11)(4,12,15,13)(6,14,16,10)(17,18)(19,25)(20,24)(21,22)(23,27)(26,32)(28,30)(29,31)(33,35)(34,38)(36,42)(37,43)(39,41)(40,45)(44,48)(46,47);(2,9)(6,16)(8,11)(10,14)(18,22)(23,25)(24,28)(31,32)(35,41)(38,42)(43,47)(45,48);(1,4,7,15)(2,10,9,14)(3,13,5,12)(6,8,16,11)(17,19,21,27)(18,23,22,25)(20,29,30,26)(24,32,28,31)(33,36,39,34)(35,42,41,38)(37,40,46,44)(43,45,47,48);(1,5,7,3)(2,11,9,8)(4,13,15,12)(6,10,16,14)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29)(33,37)(34,40)(35,43)(36,44)(38,45)(39,46)(41,47)(42,48);(1,6,7,16)(2,12,9,13)(3,14,5,10)(4,8,15,11)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,30)(29,32)(33,38,39,42)(34,41,36,35)(37,45,46,48)(40,47,44,43)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=gi,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8241548);