#8425404

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3990 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:

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

Polycyclic presentation:

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

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, 8332600);