#6441934

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437529
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=fi,d^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dhi,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349130);