#8096610

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

3881 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)(3,6,11,8)(4,9,14,16)(5,15,13,10)(17,18)(19,20)(21,29)(22,26)(23,24)(25,31)(27,32)(28,30);(1,3)(2,8)(4,13)(5,14)(6,12)(7,11)(9,15)(10,16)(17,19)(18,23)(20,26)(21,30)(22,24)(25,28)(27,31)(29,32)(34,38)(39,45)(40,46)(47,48);(1,4,7,14)(2,9,12,16)(3,13,11,5)(6,10,8,15)(17,20,22,23)(18,24,26,19)(21,28,31,32)(25,30,29,27)(33,34)(35,39)(36,40)(37,38)(41,47)(42,45)(43,48)(44,46);(1,5,7,13)(2,10,12,15)(3,14,11,4)(6,9,8,16)(17,21)(18,25)(19,27)(20,28)(22,31)(23,32)(24,30)(26,29);(1,3)(2,6)(4,13)(5,14)(7,11)(8,12)(9,10)(15,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31)(33,35,37,42)(34,39,38,45)(36,43,44,41)(40,48,46,47);(1,6)(2,11)(3,12)(4,10)(5,9)(7,8)(13,16)(14,15)(17,18)(19,23)(20,24)(21,25)(22,26)(27,32)(28,30)(29,31)(33,36,37,44)(34,40,38,46)(35,41,42,43)(39,47,45,48)

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=gh,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

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, 8003806);