#8438158

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

3970 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,d^2=h,e^2=i,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, 8345354);