#8445878

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

3948 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=g,d^2=gh,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, 8353074);