#6421875

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#2699	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1390 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,5)(4,6)(7,8)(10,14)(11,12)(17,18,23,24,22,26,19,27)(20,21,25,32,30,31,28,29);(1,3)(2,5)(4,7)(6,8)(9,10,13,14)(11,16,12,15)(17,19,22,23)(18,24,26,27)(20,25,30,28)(21,29,31,32);(2,6)(3,7)(9,11,13,12)(10,15,14,16)(17,20,22,30)(18,21,26,31)(19,28,23,25)(24,32,27,29);(2,6)(5,8)(9,11,13,12)(10,15,14,16)(17,21,19,29,22,31,23,32)(18,25,27,20,26,28,24,30);(9,12,13,11)(10,15,14,16)(17,19,22,23)(18,24,26,27)(20,25,30,28)(21,29,31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=fh,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=dfi,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6329071);