#8402285

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	#45616	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4057 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8309481);