#8750080

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

3807 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=hi,d^2=g,e^2=h,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, 8657276);