#8475172

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

4030 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^2=h,f^2=gh,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, 8382368);