#8425632

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3895 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)(3,10)(4,7)(5,12)(6,8)(9,14)(11,15)(13,16)(18,24)(20,25)(26,32)(28,30)(33,34,38,42)(35,37,41,39)(36,44,47,40)(43,46,45,48);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(17,18)(19,20)(21,25)(22,26)(23,24)(27,28)(29,30)(31,32)(33,35,38,41)(34,39,42,37)(36,45,47,43)(40,48,44,46);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,19,23,21)(18,20,24,25)(22,27,31,29)(26,28,32,30)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,20,23,25)(18,19,24,21)(22,30,31,28)(26,29,32,27)(33,36,38,47)(34,40,42,44)(35,43,41,45)(37,48,39,46);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,21,23,19)(18,25,24,20)(22,29,31,27)(26,30,32,28)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,22,23,31)(18,26,24,32)(19,27,21,29)(20,28,25,30)(33,37,38,39)(34,41,42,35)(36,46,47,48)(40,43,44,45)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=ghi,d^2=g,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,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, 8332828);