#390267

order	512
nilpotency class	3
isoclinism	isoclinic class of #30396
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1366 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fh,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 297463);