#8751359

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

3816 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)(3,12)(4,10)(5,13)(6,8)(7,9)(11,15)(14,16)(17,18,23,27)(19,25,21,24)(20,26,31,22)(28,29,30,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3,7,6)(2,8,9,12)(4,13,14,11)(5,10,15,16)(17,19,23,21)(18,24,27,25)(20,28,31,30)(22,29,26,32)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(2,9)(4,14)(8,12)(11,13)(17,19,23,21)(18,25,27,24)(20,30,31,28)(22,29,26,32)(33,34)(35,38)(36,39)(37,40)(41,43)(42,45)(44,46)(47,48);(1,4)(2,10)(3,11)(5,8)(6,13)(7,14)(9,16)(12,15)(17,20,23,31)(18,26,27,22)(19,28,21,30)(24,32,25,29)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,5)(2,11)(3,10)(4,8)(6,16)(7,15)(9,13)(12,14)(17,21,23,19)(18,24,27,25)(20,30,31,28)(22,29,26,32)(33,36,37,44)(34,39,40,46)(35,42,43,48)(38,45,41,47);(1,6,7,3)(2,8,9,12)(4,11,14,13)(5,10,15,16)(17,22,23,26)(18,20,27,31)(19,29,21,32)(24,28,25,30)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=ghi,d^2=ghi,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8658555);