#8445750

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	C₂×#120	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3965 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,c^2=gh,d^2=g,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, 8352946);