#8445896

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3954 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=g,d^2=gh,e^2=gi,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, 8353092);