#6383709

order	512
nilpotency class	2
isoclinism	isoclinic class of #2668
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator, socle	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂³×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1441 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,d^2=f,e^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ci,d^b=df,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6290905);