#8749797

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

3763 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=hi,d^2=h,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, 8656993);