#8472927

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

4261 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)(3,9)(4,8)(5,6)(7,11)(10,16)(12,13)(14,15)(20,28)(24,29)(26,32)(30,31)(33,34,38,39)(35,36,41,40)(37,42,46,45)(43,44,48,47);(1,3,7,4)(2,8,11,9)(5,14,16,12)(6,13,10,15)(17,18)(19,23)(20,29)(21,25)(22,27)(24,28)(26,30)(31,32)(34,39)(36,40)(42,45)(44,47);(1,3,7,4)(2,9,11,8)(5,12,16,14)(6,13,10,15)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31)(33,35,38,41)(34,40,39,36)(37,43,46,48)(42,47,45,44);(1,4,7,3)(2,9,11,8)(5,14,16,12)(6,13,10,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,32)(29,30)(33,34)(35,40)(36,41)(37,45)(38,39)(42,46)(43,44)(47,48);(1,5)(2,10)(3,12)(4,14)(6,11)(7,16)(8,13)(9,15)(17,20)(18,24)(19,26)(21,28)(22,30)(23,31)(25,29)(27,32)(33,36,38,40)(34,41,39,35)(37,44,46,47)(42,48,45,43);(1,6,7,10)(2,5,11,16)(3,13,4,15)(8,14,9,12)(17,18)(19,22)(20,24)(21,25)(23,27)(26,30)(28,29)(31,32)(33,37,38,46)(34,42,39,45)(35,43,41,48)(36,44,40,47)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=g,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8380123);