#8438078

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4190 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,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, 8345274);