#8445881

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	#2430	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3977 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,d^2=h,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, 8353077);