#8095023

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	#2670	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3915 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=i,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002219);