#8459208

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

3970 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)(3,8,4,7)(5,9,14,13)(11,16,12,15)(17,18,23,24)(19,25,20,22)(21,29,31,26)(27,30,28,32)(36,44)(40,47)(43,48)(45,46);(1,3,6,4)(2,7,10,8)(5,11,14,12)(9,15,13,16)(18,24)(19,20)(26,29)(27,28)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(17,19,23,20)(18,22,24,25)(21,27,31,28)(26,30,29,32)(33,35)(34,37)(36,43)(38,41)(39,42)(40,45)(44,48)(46,47);(1,2)(3,7)(4,8)(5,13)(6,10)(9,14)(11,16)(12,15)(17,20,23,19)(18,25,24,22)(21,28,31,27)(26,32,29,30)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,48)(45,47);(1,4,6,3)(2,8,10,7)(5,12,14,11)(9,16,13,15)(17,21,23,31)(18,26,24,29)(19,27,20,28)(22,30,25,32)(33,36,38,44)(34,40,42,47)(35,43,41,48)(37,45,39,46);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,16)(10,13)(17,22,23,25)(18,19,24,20)(21,30,31,32)(26,27,29,28)(33,37,38,39)(34,41,42,35)(36,45,44,46)(40,48,47,43)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,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, 8366404);