#8425401

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3916 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gi,d^2=h,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, 8332597);