#7045337

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996335
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1398 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=fgi,d^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6952533);