#8445872

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

3871 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=h,e^2=i,f^2=g,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, 8353068);